Hankel_transform.ipynb

{
 "metadata": {
  "name": "Hankel_transform"
 },
 "nbformat": 3,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "input": "%load_ext sympyprt",
     "language": "python",
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "input": "from sympy import *",
     "language": "python",
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "input": "r = Symbol(\"r\")\nk = Symbol(\"k\")\n\nnu = Symbol(\"nu\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "source": "Zero-th order"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "1/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAlBAMAAACJ5mVtAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEKtEMol2VCK73Zlm72LS\nT58AAABKSURBVAgdY2BgVGZgYGAJew4kGRjIITkiX20GayaSeAcCRKpFV8ae1pbGwCDTl6DJwGBY\nyyAAlN8EVrMGRDK+AJHsF0AkWwCQAADSdRbnwMlO3gAAAABJRU5ErkJggg==\n",
       "prompt_number": 4,
       "text": "\n1\n\u2500\nr"
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAlBAMAAABmJA5TAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAAAaElEQVQYGWNgYFRmAAKWsO8gioGBIooj8tdmiDGkkf9B4AN+PTUfwfLsF8AUqwKY\n4nkApuQbwJQtQ+kBICOITXwBkDr6AEgwMP5MAVHMF+QdgBSrAs+DAgYGngdchg8YGLga2LcxMAAA\nX08hl99B2lwAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": "\n1\n\u2500\nk"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAlBAMAAACJ5mVtAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEKtEMol2VCK73Zlm72LS\nT58AAABKSURBVAgdY2BgVGZgYGAJew4kGRjIITkiX20GayaSeAcCRKpFV8ae1pbGwCDTl6DJwGBY\nyyAAlN8EVrMGRDK+AJHsF0AkWwCQAADSdRbnwMlO3gAAAABJRU5ErkJggg==\n",
       "prompt_number": 6,
       "text": "\n1\n\u2500\nr"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "source": "1b"
    },
    {
     "cell_type": "code",
     "input": "m = Symbol(\"m\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "input": "1/r**m",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAANBAMAAADVpBoRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAb0lEQVQIHWNggAOn5KC1QQpwLpAhsInhI8sEqEjUqlWrAlgmME/gKEBWw1jAuoAr\nAFmES4FNgNOJgYFRyUgJIs5mwOfApcvA4GqvkI6skoGhRZLBAVWEYRkan4FhK7oIy2d0EUaYK+ES\nTAfgTAgDADWBEV+IwoMIAAAAAElFTkSuQmCC\n",
       "prompt_number": 8,
       "text": "\n -m\nr  "
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAAxBAMAAACYFXDMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAmAARgTjqeq1afwD/HxSMtVEZOUAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": "\n -m + 1  m - 2  \u239b  m    \u239e\n2      \u22c5k     \u22c5\u0393\u239c- \u2500 + 1\u239f\n                \u239d  2    \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            \u239bm\u239e          \n           \u0393\u239c\u2500\u239f          \n            \u239d2\u23a0          "
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAANBAMAAADVpBoRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAb0lEQVQIHWNggAOn5KC1QQpwLpAhsInhI8sEqEjUqlWrAlgmME/gKEBWw1jAuoAr\nAFmES4FNgNOJgYFRyUgJIs5mwOfApcvA4GqvkI6skoGhRZLBAVWEYRkan4FhK7oIy2d0EUaYK+ES\nTAfgTAgDADWBEV+IwoMIAAAAAElFTkSuQmCC\n",
       "prompt_number": 10,
       "text": "\n -m\nr  "
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAIBAMAAAA2IaO4AAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAVHYiEGaru0Tvmd2JcbBxRwAA\nACtJREFUCB1jYFQyUmJwtVVIZ2iRZHBgYFjGAARbgZjlMpBgnAAkmA4wMAAAeHEF42gIMlAAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 11,
       "text": "r"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "input": "HankelTransform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANQAAAASBAMAAADRQe2CAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nHo+QFKDLwj8oeS2ngPx3mQAAAABJRU5ErkJggg==\n",
       "prompt_number": 12,
       "text": "HankelTransform(r, r, k, 0)"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "input": "_.as_integral",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAA9BAMAAABoypv0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\neWFVSYrCqhAVhbX35B+klKCxeh35SwAAAABJRU5ErkJggg==\n",
       "prompt_number": 13,
       "text": "\n\u221e                      \n\u2320                      \n\u23ae  2                   \n\u23ae r \u22c5besselj(0, k\u22c5r) dr\n\u2321                      \n0                      "
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "r**3",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABAAAAAQBAMAAADt3eJSAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVHYiEGaru0Tvmd2JMvIc\n2k8AAABcSURBVAgdY2AAgcawC2CaYS7HBAjDgOMBhMHAFQBh8B5zgIpwTAEzGCcwPAUzeA+wvGVg\nVDJSYhAKE2BwtVNIBwu3SDI4gBkMyyAUA8NWKIPlMZQBNAMCmA5AaADF3Q+dvcGTxgAAAABJRU5E\nrkJggg==\n",
       "prompt_number": 14,
       "text": "\n 3\nr "
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "input": "m = Symbol(\"m\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "input": "r**m",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAANBAMAAAC0vDhTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAZElEQVQIHWNgAAKn5KC1QQogFgODwCaGjywTIGyWCcwTOAogbMYC1gVcARA2lwKb\nAKcTAwOjkpESmwGfA5cuA4OrvUI6RJaBoUWSwQHGZlgGZzEwbEWwWT4j2IxQ+0AiTAfg4gCFsA63\nEURDmAAAAABJRU5ErkJggg==\n",
       "prompt_number": 17,
       "text": "\n m\nr "
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "input": "h = hankel_transform(_, r, k, 0)\nh",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALcAAAAxBAMAAABqqSVpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTjHby8MRScIZFq96+N9f9EsM2/hBWQAAAABJRU5ErkJggg==\n",
       "prompt_number": 18,
       "text": "\n m + 1  -m - 2  \u239bm    \u239e\n2     \u22c5k      \u22c5\u0393\u239c\u2500 + 1\u239f\n                \u239d2    \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          \u239b-m\u239e         \n         \u0393\u239c\u2500\u2500\u239f         \n          \u239d2 \u23a0         "
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "input": "h.subs(m, 3)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAAlBAMAAAC9n6RxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2MmbduyIQie+r\nzUQwcvfeAAAAv0lEQVQYGWNgYGAQMnIAkkDAmsCQAmHxGzCoQ1j9AQz2ENb8AIb4CWAm2wIGfQcw\niyWTA8piaA6EqgNKqIElGRgDGDZCWOwHmA9AWByrFkMYVCH/wwBu07j+Q93AwDwTpEr2JwMDM1g5\n4wYgK1QFyOY6wMDAyDAbyOINAElxOzAwzG9gYPAGs14zSF2YxeAGFF/DPDGBz1gTyLoFVgdkcH43\nAZIgwLdhvgOYATSON0AAzOQNYHsJUcnWwKgMFgIALM0pUBFj3QQAAAAASUVORK5CYII=\n",
       "prompt_number": 19,
       "text": "\n9 \n\u2500\u2500\n 5\nk "
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(h, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAANBAMAAAC0vDhTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAZElEQVQIHWNgAAKn5KC1QQogFgODwCaGjywTIGyWCcwTOAogbMYC1gVcARA2lwKb\nAKcTAwOjkpESmwGfA5cuA4OrvUI6RJaBoUWSwQHGZlgGZzEwbEWwWT4j2IxQ+0AiTAfg4gCFsA63\nEURDmAAAAABJRU5ErkJggg==\n",
       "prompt_number": 20,
       "text": "\n m\nr "
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "input": "# Only the general case with symbolic 'm' does work!",
     "language": "python",
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "input": "1/sqrt(r**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAqBAMAAADi/wAlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAABX0lEQVQ4Ee2UP0vDQBjGn2tiL2kQpYs6taNjd5cg2FXBLxAESzed3MQquFe3gmDX\n4tJBBDfxE+ggON4gKBWkUvHPFN+73KUhgdBBEMQbnvd5f89LuLuQAOlVrqVJtl/Z2s7CDJn5H/qx\nKzg8PZvgvWRewZ8H4QTrly5hNW9rZk9HxuRUu6/C+oafHkogtyVDFti3qaEkWlaZPcQoNZREi1Hm\ns89oevwVGwQ4Vf0Ab00ZLnRPRSOg4FMn13NUuIiqVI2AA2oaj11YAZMcXJAAO81GECE5cA64gddF\nff5BhuCCBOzpvdBWyKL/EWsDA3h9JwyHMgUXJLDmbphG7hWmfOAOJXOky85Jr3NMU16VRC3rDQt0\nvhEqswaBC2VLUZG+h3XAfsWAq0QKFyRApaWKlL192hLunetijLhQdjcGmL6oUbP0Um7GjAtlN2OA\n4tfYa2eZM8QJ+4htjvFzMhV9A217jHtwiRTvAAAAAElFTkSuQmCC\n",
       "prompt_number": 23,
       "text": "\n     1      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         1/2\n\u239b 2    2\u239e   \n\u239da  + r \u23a0   "
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAnBAMAAABUEX/sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfwA/WSikEU0aiAAAAABJRU5ErkJggg==\n",
       "prompt_number": 24,
       "text": "\n-sinh(a\u22c5k) + cosh(a\u22c5k)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          k           "
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "source": "6"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "input": "1/(r**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAmBAMAAACMti74AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA/0lEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IFyL7OkqWAY\n4pbkABUTY/AVQJNmTGC5CBXyZ+A3QJNl+cDwGSrEwiDfgCbL4MD4Ay60FcJyhwswMHAHwDg8GyCs\nGpgAkD4NZ5sxQFyFJMucwAiUZsupYmBJ7HEAq4TIsqe1pTG4SRwBCk1j2MXQ//8/WJIBIivTn6DJ\n8f//BwYGngUMuyEyIBIia1gLtYiBN4BhOVSWe+bMVzNnKgB5m6AiDP0GjB9hbJhehjUwEX8BngUF\nMA7UZEQ48Db0KSSgybJfgAkw7zmo1gDjQPWyBcAFUBjmKDwGAOi7VkS1Y1DSAAAAAElFTkSuQmCC\n",
       "prompt_number": 27,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2\na  + r "
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALEAAAAuCAMAAABQxwhhAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 28,
       "text": "\n        \u239b        \u2502  2  2\u239e\n\u256d\u2500\u256e2, 1 \u239c 0      \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c        \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d0, 0  0 \u2502   4  \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            2            "
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALEAAAAuCAMAAABQxwhhAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 29,
       "text": "\n        \u239b        \u2502  2  2\u239e\n\u256d\u2500\u256e2, 1 \u239c 0      \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c        \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d0, 0  0 \u2502   4  \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            2            "
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "input": "# Should be a besselk_0(k*z)",
     "language": "python",
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAmBAMAAACMti74AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA/0lEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IFyL7OkqWAY\n4pbkABUTY/AVQJNmTGC5CBXyZ+A3QJNl+cDwGSrEwiDfgCbL4MD4Ay60FcJyhwswMHAHwDg8GyCs\nGpgAkD4NZ5sxQFyFJMucwAiUZsupYmBJ7HEAq4TIsqe1pTG4SRwBCk1j2MXQ//8/WJIBIivTn6DJ\n8f//BwYGngUMuyEyIBIia1gLtYiBN4BhOVSWe+bMVzNnKgB5m6AiDP0GjB9hbJhehjUwEX8BngUF\nMA7UZEQ48Db0KSSgybJfgAkw7zmo1gDjQPWyBcAFUBjmKDwGAOi7VkS1Y1DSAAAAAElFTkSuQmCC\n",
       "prompt_number": 31,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2\na  + r "
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "source": "6b"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "input": "1/sqrt(r**4+a**4)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAqBAMAAADi/wAlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAABU0lEQVQ4EWNgQAdCBugimHzX/AJMQQwR/lFFVAuC1vmriYgXjCgY9gL/iQADFAj+\n+JwGc9NEGAMPzbIBIsnqgKEIIcTZAJG0xVSEEHKCqGF8g6EISUgDooiNB6KIBZGLYUIMDBwKEEUV\nUEXsDyB8IAkTYmBgcgByGRgYCzAUwYUYGJqAKtKeL+BLy90MVs3+AEyVZaTlQYRACrYwMHAmcC9g\nYOByAMuyPwBRjC++MU0ACzEDyyPGCQwMrxi4NzAw5iuDZBnYHwAJBmbxC4xQIc4DDKDwusrABfPS\nzplzVs6cBFTFrQAkwID5C4Mk0H+fGeQFYEIM7A/ATC4IBWKvZAhmYGD5yPCKHSwDItgfAAkGBvkG\nMAUi6huBTmK4wXGIDS7E/gDMrIULMPBuMwByrN8JZcDF2B+AmalwAQa2nwg2lMUM8wNchvE7nImH\n4YBHDiwFAMitgqE9teueAAAAAElFTkSuQmCC\n",
       "prompt_number": 33,
       "text": "\n     1      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         1/2\n\u239b 4    4\u239e   \n\u239da  + r \u23a0   "
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANwAAABGCAMAAACQeQqWAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 34,
       "text": "\n        \u239b                  \u2502  4  4\u239e\n\u256d\u2500\u256e3, 1 \u239c   1/2            \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c                  \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 5 \u239d0, 0, 1/2  0, 1/2 \u2502  256 \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  ___              \n              4\u22c5\u2572\u2571 \u03c0               "
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "input": "# Should be a product of K_0 and J_0",
     "language": "python",
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAA4BAMAAAB+santAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYv3bVSgS3LYkUIE8BZfhOxkU7AfbR6yqYfgX+AAAAABJRU5ErkJggg==\n",
       "prompt_number": 36,
       "text": "\n        1        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         ________\n        \u2571  4     \n 2     \u2571  a      \nr \u22c5   \u2571   \u2500\u2500 + 1 \n     \u2571     4     \n   \u2572\u2571     r      "
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "source": "6c"
    },
    {
     "cell_type": "code",
     "input": "f = 1/(r*(r+a))\nf",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAApBAMAAACci4l8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAABSUlEQVQ4EWNgQANCBmgCGFzX/AIMMXQB/lEl6EEC5hMOl9b5qwnGAFajh5vgf0Lg\nA/19zIPVShRRDaxKGDcghHlw5I0yhBIuBJOBBUk5nwBcAkk1A/sBuDADG4KtjBBFUcLygIGBPa0t\nDSj9AYjTji8AkkChA2CqLCMtASwh05+gycDACFTCmcC9ACwHUcJ44hvTBAaGiwwMhrUMQBdxXGBg\nOMXAvQFJCbP4BUYgdwUQbwIJcyxgYLjKwFUAZO6cOW/lzElABrcCkGBYAsRrQAzGCwwcnxnkgeYB\nAdQtXAdAHKBFjD9ADKBzWT4ynGIHs6FK5BsgEgzsQFcAwRMGhhscj9jAbKiSWhCHZQIDA1sAWBjo\nceszQhnISlJBHJ4DYCEQIQVnMTAwFyA4SBHAaoAQRmaZInEUkdgIJqMCgs0ghsRGMNkEGBgATE5h\nYX4zDfEAAAAASUVORK5CYII=\n",
       "prompt_number": 37,
       "text": "\n    1    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\nr\u22c5(a + r)"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAABBCAMAAAC+RyUnAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 38,
       "text": "\n        \u239b             \u2502  2  2\u239e\n\u256d\u2500\u256e3, 2 \u239c 1/2, 0      \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f2, 4 \u239d1/2, 0, 0  0 \u2502   4  \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n             2\u22c5\u03c0              "
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "input": "# Expressible in Kelvin functions",
     "language": "python",
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAApBAMAAACci4l8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAABSUlEQVQ4EWNgQANCBmgCGFzX/AIMMXQB/lEl6EEC5hMOl9b5qwnGAFajh5vgf0Lg\nA/19zIPVShRRDaxKGDcghHlw5I0yhBIuBJOBBUk5nwBcAkk1A/sBuDADG4KtjBBFUcLygIGBPa0t\nDSj9AYjTji8AkkChA2CqLCMtASwh05+gycDACFTCmcC9ACwHUcJ44hvTBAaGiwwMhrUMQBdxXGBg\nOMXAvQFJCbP4BUYgdwUQbwIJcyxgYLjKwFUAZO6cOW/lzElABrcCkGBYAsRrQAzGCwwcnxnkgeYB\nAdQtXAdAHKBFjD9ADKBzWT4ynGIHs6FK5BsgEgzsQFcAwRMGhhscj9jAbKiSWhCHZQIDA1sAWBjo\nceszQhnISlJBHJ4DYCEQIQVnMTAwFyA4SBHAaoAQRmaZInEUkdgIJqMCgs0ghsRGMNkEGBgATE5h\nYX4zDfEAAAAASUVORK5CYII=\n",
       "prompt_number": 40,
       "text": "\n    1    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\nr\u22c5(a + r)"
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOEAAABBCAMAAAAQcTOmAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAEImrmSJ2\n3UTvVLvNMmZAoe8m1AUAAAZ1SURBVGgF7VrbmqsqDAZU6nlt3/9ldwIEwslaxs7qrG+8qBCTkB9D\nCLFC3HEtwx1ayjrkVqa/izqMueZZTW9EuKl8xCcUWTDyiYh/PKzSt1njwxCKfmfGvdSUZYDi0xCK\nZX4JV2BeK4IpwrnfVYU1KLvYavBSIeSjbd2ormJVgnBcgW/5wmLgwzQhFOOD67ja3qbiIgTxBOGK\ny2BHmDdcbQjFohvGXqpRLUYoD/TQ8ajNx2tDNyLcGoavyoz66Hjw2g500O24ZyNrRCjq76M6w0tt\nFaYSo8FmcabPXu+3IhyP8ljjvHdu7jetqYm88vI7+QyEYuJ+FdBOmxgn05WwVHls0ZYcWKl1uIv6\n4J/opRanJzY3/vznRF9NDftyqINtxL3eDd+ysdWOsF51UnjbGGnmhqXusEQ356WF1DBxMpCKKPVQ\nRykPOKs8/MZpzY7Gpk6kF4kmUt++W8QBG4xLnCyjmIkmI8N9U12I8n3YH+qvJIy0PewSNjv+eveO\nnyIMTjav1l8CxeB5BOuFVF2v1DAY0ug39r0PyFVtGcKyAy63QTids9r17VlbilAEJ5udcYGChneL\nN18/zHyraTashwtCI7iBf59Led2iDqa3HL78QC0Nv1tkCEGbczKWTjK3036/WNyJQR4S1eG2MILV\nm5bDDnd7PU4DDen9ZoTkZCZNNHYSBTszRZGeGgJAzHrvwL/6RQxmE7Dw4HdiTu2J1CC9sw9M9OTr\n95N3SE4WAiJRzLCUc4xHKUfNXga5bslkr/eutccHcQhZaqj1PuPRwTsZ7uCzgmXkKUbB4ILpUty3\nMtSV0IuqYr1G+Y0//h2STtyLekAYnAyWoZYAOlAMr3Rv5bAhRPcry1VnH2GcXpoPGobdE73syS3N\nFOGOr69nUV6IVWUuZ4Z23nlQCDkiqcQ68mkiX0mhrvCQPrrnMjZrC6mhCQgPvh7ksWgfEUkR3g8b\nOyaHcDvxQ/BElr+h05dCN1d+jSeWKMsk73BAQyDqM1FYhuVjgUPYu93ShtQ4A1N+alIvfY4wO/kz\nm+rNXG+C0JxSwVPZS8TdEPYLRnH6KYauxjvHDpGGDAyZRjhpuItWLfVzS+hJuF/hCdy2lcsEhHa+\ngWOAUwALhbgbPkaZr0Uf/3XXa0iykCNkYNCR++oRCs9ds8TS+W9uLX9abucyHqGb77Hfd9lptvNi\nXW3uGWSnunxYAEh+/9SCI4w1pJaU6ocpTxlTTM1lCGE037FQpWeWbOkZZWCQoDCEUxxpE0uK9cOE\npzRWRstlCGE035lciTD6ZC1+ShmYhJXLEC4hUUf+xJJi/TDhiYep9HIZhzCe74p0TN595h3RfQa2\na62n3h/yoiIGS6GMcKl+mPJEo1Q6JRmLMJnvinxMxsQnv1wGhmcLuNyJFpv1mgA8tLtlpX64r2Y/\nigqOqPHsYjIWYTLfZ7L+WbE4SBkYnC1w6zg6/w5ltOV7LbZxXl1TqGyoLIpEE3WDDK3DaL6J7fSe\nhP+EN99cWE0gpFBOqoCQ8ZiyjTE5GSTtFmWo1hbNdypZ6tcCjeON9wYkqqJXW3brpbX6IRZaZR5C\nQHRY/JUMGGTCO7RjXf49Mxg2UJ73EQq2ySbDnNYPTRZR+y6XKKIuk2lGuJ6d2Wmg6P4I0xxnr8BV\nqB96Hg3Hs4thpiTTirD6mSUCFXXMudpQQvaa1Q+JwDLcRQ3860CkM+4EvUymFSGrtMWjnPQeFH5C\n9rq5Awqcp2z90BMCz3T5c3BRphGhzz1PAGWPqCoJ+1/IXgm15yYC4/HPnjVKMo0IWYHx2ajs+ULW\nA42yV0aynIxAPEzF02Ym04Zw8GXtpyNyBvb9n7LXrH7ICMTDVTxr5zJtCFu/KZgjBBrps9fsYB0I\nnucZKva8INOEUL28U5ARsy3qXKkfXuEhtXQvybQgnOOjHqm/dJ9xmVH2eiZxhSeVL8o0IBy+ADC1\n6bf/CTNA+fw/e/+ESf614XcGfmfgdwZumoFd9c1/3L3JhPeqwe8M23nx670GvF37hCUQ83nr7UP9\npQFMLVJFHzb/kiVvHXaB4rdcXPZUr9y91Ya3Kpf4TbbbpJLjmFcr3zr0Nylf4FCzSTHsgn/f/KbB\nv2MY7c7dasA/X/2D1wwuanwTCsQnHwh+LnKswJi/F4zwAosfMX4uNmP5tuz7rvEzG5YkfW35h6Pi\n5k9mj0CED3DVwh8nOPOt7f8Byhc/SflMHfQAAAAASUVORK5CYII=\n",
       "prompt_number": 41,
       "text": "\n        \u239b 1/2, 0       \u2502  2  2\u239e\n\u256d\u2500\u256e3, 2 \u239c              \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c        \u03bd  -\u03bd \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f2, 4 \u239c1/2, 0, \u2500  \u2500\u2500 \u2502   4  \u239f\n        \u239d        2  2  \u2502      \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n              2\u22c5\u03c0              "
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "source": "6d"
    },
    {
     "cell_type": "code",
     "input": "f = 1/(1+r**4)\nf",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAmBAMAAABwowUyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAApklEQVQoFWNgQAAhAwQbmeWaX4DMRWLzD3OJ1vmrcYQJUiAMWuZ/HIBsB4fg0Ml8\nBYeEzBMcEgmoEozKUHXsDVAJd7AAS9h3qIQ4A1SiBioAkziWdhsSRcgS7GltaQwMSyBKkSVk+hM0\nGcQ+YeowrGUQgJrMwICsg2ETTJh75sxXM2cqgLhgy9fAJIA0sg7GHzgk2C/gkGALwJDgiPy1GUkQ\nxDRH8AG1MkUB93THewAAAABJRU5ErkJggg==\n",
       "prompt_number": 42,
       "text": "\n  1   \n\u2500\u2500\u2500\u2500\u2500\u2500\n 4    \nr  + 1"
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANoAAAAuCAMAAAB5wPfmAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 43,
       "text": "\n        \u239b                    \u2502   4\u239e\n\u256d\u2500\u256e3, 1 \u239c    1/2             \u2502  k \u239f\n\u2502\u2576\u2510     \u239c                    \u2502 \u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 5 \u239d1/2, 0, 1/2  0, 1/2 \u2502 256\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                 4                 "
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAmBAMAAABwowUyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAApklEQVQoFWNgQAAhAwQbmeWaX4DMRWLzD3OJ1vmrcYQJUiAMWuZ/HIBsB4fg0Ml8\nBYeEzBMcEgmoEozKUHXsDVAJd7AAS9h3qIQ4A1SiBioAkziWdhsSRcgS7GltaQwMSyBKkSVk+hM0\nGcQ+YeowrGUQgJrMwICsg2ETTJh75sxXM2cqgLhgy9fAJIA0sg7GHzgk2C/gkGALwJDgiPy1GUkQ\nxDRH8AG1MkUB93THewAAAABJRU5ErkJggg==\n",
       "prompt_number": 44,
       "text": "\n  1   \n\u2500\u2500\u2500\u2500\u2500\u2500\n 4    \nr  + 1"
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(f, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "source": "6e"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "input": "f = 1 / (a**4+r**4)\nf",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAmBAMAAACMti74AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA70lEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IEaXg7BNITV\nASbGfAXGQtC2cFmZJwhRKIvxHFw2AVOWjQ8my94AlXVHGFEBlxVngMrWwGUZC+Cyx9JuG4DFEbI8\nabmglMCWUwUkl0A0QWTZ09rSGBi4HIBi0xh2MTCIfULWK9OfoMnAmA/MkDwLGHZD9IFIiF7DWgYB\niBhvAMNyqCz3zJmvZs5UAPI2QUUY+g0YP8LYML0Ma2Ai/gI8CwpgHKjJjD9gArwNfQoJMA5Ulv0C\nTIB5z0G1BhgHKssWABdAYZij8BgAoNpTvsD7+gsAAAAASUVORK5CYII=\n",
       "prompt_number": 47,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 4    4\na  + r "
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAN4AAABCCAMAAAAPHZi9AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 48,
       "text": "\n        \u239b                    \u2502  4  4\u239e\n\u256d\u2500\u256e3, 1 \u239c    1/2             \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c                    \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 5 \u239d1/2, 0, 1/2  0, 1/2 \u2502  256 \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    2                \n                 4\u22c5a                 "
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAmBAMAAACMti74AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA70lEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IEaXg7BNITV\nASbGfAXGQtC2cFmZJwhRKIvxHFw2AVOWjQ8my94AlXVHGFEBlxVngMrWwGUZC+Cyx9JuG4DFEbI8\nabmglMCWUwUkl0A0QWTZ09rSGBi4HIBi0xh2MTCIfULWK9OfoMnAmA/MkDwLGHZD9IFIiF7DWgYB\niBhvAMNyqCz3zJmvZs5UAPI2QUUY+g0YP8LYML0Ma2Ai/gI8CwpgHKjJjD9gArwNfQoJMA5Ulv0C\nTIB5z0G1BhgHKssWABdAYZij8BgAoNpTvsD7+gsAAAAASUVORK5CYII=\n",
       "prompt_number": 49,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 4    4\na  + r "
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAEG8AAABJCAMAAACqGEyyAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nW47dYwo2AAAAAElFTkSuQmCC\n",
       "prompt_number": 50,
       "text": "\n\u239b       \u2148\u22c5\u03c0\u22c5(2\u22c5\u03bd + 1)                                                         \n\u239c       \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                 \u239b              1        \n\u239c 3  3        4          \u239b\u03c0\u22c5\u03bd\u239e    \u239b\u03c0\u22c5\u03bd\u239e  \u239b\u03bd   1\u239e  \u250c\u2500  \u239c                       \n\u239ca \u22c5k \u22c5\u212f             \u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 - \u2500\u239f\u22c5 \u251c\u2500  \u239c  \u03bd      \u03bd        \u03bd   3\n\u239c                        \u239d 4 \u23a0    \u239d 4 \u23a0  \u239d4   2\u23a0 1\u2575 4 \u239c- \u2500 + 1, \u2500 + 1, - \u2500 + \u2500\n\u239c                                                     \u239d  4      4        4   2\n\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239d                                              16                             \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n        \u2502   4  4\u239e            \u2148\u22c5\u03c0\u22c5(\u03bd + 1)                 \u239b        \u2148\u22c5\u03c0\u239e        \n        \u2502 -a \u22c5k \u239f            \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                 \u239c        \u2500\u2500\u2500\u239f        \n  \u03bd   3 \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u239f                 4         \u239b\u03c0\u22c5\u03bd\u239e        \u239c         4 \u239f  \u239b\u03bd   3\n, \u2500 + \u2500 \u2502  256  \u239f   \u03c0\u22c5a\u22c5k\u22c5\u03bd\u22c5\u212f           \u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5besseli\u239d\u03bd, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\n  4   2 \u2502       \u23a0                           \u239d 4 \u23a0                       \u239d4   2\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                 8                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n             \u2148\u22c5\u03c0\u22c5(\u03bd + 1)                 \u239b        \u2148\u22c5\u03c0\u239e                       \u2148\n             \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                 \u239c        \u2500\u2500\u2500\u239f                       \u2500\n\u239e                 4         \u239b\u03c0\u22c5\u03bd\u239e        \u239c         4 \u239f  \u239b\u03bd   3\u239e               \n\u239f   \u03c0\u22c5a\u22c5k\u22c5\u03bd\u22c5\u212f           \u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5besselj\u239d\u03bd, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f   \u2148\u22c5\u03c0\u22c5a\u22c5k\u22c5\u03bd\u22c5\u212f \n\u23a0                           \u239d 4 \u23a0                       \u239d4   2\u23a0               \n\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                 8                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n\u22c5\u03c0\u22c5(\u03bd + 1)                 \u239b            \u2148\u22c5\u03c0\u239e                       \u2148\u22c5\u03c0\u22c5(\u03bd + 1)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                 \u239c            \u2500\u2500\u2500\u239f                       \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    4         \u239b\u03c0\u22c5\u03bd\u239e        \u239c             4 \u239f  \u239b\u03bd   3\u239e                   4     \n          \u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besseli\u239d\u03bd + 2, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f   \u2148\u22c5\u03c0\u22c5a\u22c5k\u22c5\u03bd\u22c5\u212f           \n              \u239d 4 \u23a0                           \u239d4   2\u23a0                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    8                                                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                     3        \u239b\u03c0\u22c5\u03bd\u239e    \u239b\u03c0\u22c5\u03bd\u239e  \n                                                    a \u22c5k\u22c5\u03bd\u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\n                                                              \u239d 4 \u23a0    \u239d 4 \u23a0  \n\n                                                                              \n                 \u239b            \u2148\u22c5\u03c0\u239e                    \u2148\u22c5\u03c0\u22c5\u03bd                 \u239b \n                 \u239c            \u2500\u2500\u2500\u239f                    \u2500\u2500\u2500\u2500\u2500                 \u239c \n    \u239b\u03c0\u22c5\u03bd\u239e        \u239c             4 \u239f  \u239b\u03bd   3\u239e        2    4      \u239b\u03c0\u22c5\u03bd\u239e        \u239c \n\u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besselj\u239d\u03bd + 2, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f   \u2148\u22c5\u03c0\u22c5\u03bd \u22c5\u212f     \u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besseli\u239d\u03bd\n    \u239d 4 \u23a0                           \u239d4   2\u23a0                    \u239d 4 \u23a0          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          8                                                              4    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u239b\u03bd   3\u239e                                                                       \n\u239c\u2500 + \u2500\u239f                                                                       \n\u239d4   2\u23a0                                                                       \n\n                                                                              \n           \u2148\u22c5\u03c0\u239e                    \u2148\u22c5\u03c0\u22c5\u03bd                 \u239b            \u2148\u22c5\u03c0\u239e    \n           \u2500\u2500\u2500\u239f                    \u2500\u2500\u2500\u2500\u2500                 \u239c            \u2500\u2500\u2500\u239f    \n            4 \u239f  \u239b\u03bd   3\u239e        2    4      \u239b\u03c0\u22c5\u03bd\u239e        \u239c             4 \u239f  \u239b\u03bd\n + 1, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f   \u2148\u22c5\u03c0\u22c5\u03bd \u22c5\u212f     \u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besselj\u239d\u03bd + 1, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500\n                 \u239d4   2\u23a0                    \u239d 4 \u23a0                           \u239d4\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                      4                       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n               \u2148\u22c5\u03c0\u22c5\u03bd                 \u239b            \u2148\u22c5\u03c0\u239e                   \u2148\u22c5\u03c0\u22c5\u03bd\n               \u2500\u2500\u2500\u2500\u2500                 \u239c            \u2500\u2500\u2500\u239f                   \u2500\u2500\u2500\u2500\u2500\n   3\u239e            4      \u239b\u03c0\u22c5\u03bd\u239e        \u239c             4 \u239f  \u239b\u03bd   3\u239e            4  \n + \u2500\u239f   \u2148\u22c5\u03c0\u22c5\u03bd\u22c5\u212f     \u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besseli\u239d\u03bd + 1, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f   \u2148\u22c5\u03c0\u22c5\u03bd\u22c5\u212f     \n   2\u23a0                   \u239d 4 \u23a0                           \u239d4   2\u23a0               \n\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                   4                                          \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                           \u239e                \n                 \u239b            \u2148\u22c5\u03c0\u239e         \u239f                \n                 \u239c            \u2500\u2500\u2500\u239f         \u239f                \n    \u239b\u03c0\u22c5\u03bd\u239e        \u239c             4 \u239f  \u239b\u03bd   3\u239e\u239f  \u2148\u22c5\u03c0\u22c5(-2\u22c5\u03bd - 1)\n\u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5besselj\u239d\u03bd + 1, a\u22c5k\u22c5\u212f   \u23a0\u22c5\u0393\u239c\u2500 + \u2500\u239f\u239f  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    \u239d 4 \u23a0                           \u239d4   2\u23a0\u239f        4       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f              \n               4                           \u23a0                \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                            \n                                                            \n                                                            "
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "source": "7"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "input": "exp(I*a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAmBAMAAAC4z+/kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAoklEQVQoFWNggAMlVU/VVQ/gXAYGZoX5XLwLoCLGQDDBf8IehulIKhgY1jKoMSxF\nEeljuCFgBBZhVDQB08yCygtCwKwwBicUtQxsBQwpqCL8gYoOqCL+Dah8Bob5DgycqGKsCxgkUEU4\nDYUcUEUGlvcfDXwYWOeQYDuXkpMSqvKJ/gpPUUU6lzNMQBVhsEPjMzBcRBfh/IIuwlWALsKUABcB\nAEspKfiBHXxPAAAAAElFTkSuQmCC\n",
       "prompt_number": 53,
       "text": "\n \u2148\u22c5a\u22c5r\n\u212f     \n\u2500\u2500\u2500\u2500\u2500\u2500\n  r   "
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "input": "h = hankel_transform(_, r, k, 0)\nh",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANsAAABXCAMAAAC6L8DRAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 54,
       "text": "\n\u23a7        \u2148\u22c5\u03c0                        \n\u23aa        \u2500\u2500\u2500                        \n\u23aa         2                         \n\u23aa       \u212f                 2 \u25021 \u2502    \n\u23aa \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   for a \u22c5\u2502\u2500\u2500\u2502 > 1\n\u23aa         ________          \u2502 2\u2502    \n\u23aa        \u2571  2               \u2502k \u2502    \n\u23aa       \u2571  a                        \n\u23aa k\u22c5   \u2571   \u2500\u2500 - 1                   \n\u23aa     \u2571     2                       \n\u23a8   \u2572\u2571     k                        \n\u23aa                                   \n\u23aa        1                          \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n\u23aa        __________                 \n\u23aa       \u2571    2                      \n\u23aa      \u2571    a                       \n\u23aak\u22c5   \u2571   - \u2500\u2500 + 1                  \n\u23aa    \u2571       2                      \n\u23aa  \u2572\u2571       k                       \n\u23a9                                   "
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "input": "# Could be correct but should be simplified to nicer form\n# - Put the k into the roots\n# - simplify exp to I",
     "language": "python",
     "outputs": [],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "input": "h.args[0].args[0]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAABABAMAAACpTDdgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 56,
       "text": "\n       \u2148\u22c5\u03c0      \n       \u2500\u2500\u2500      \n        2       \n      \u212f         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        ________\n       \u2571  2     \n      \u2571  a      \nk\u22c5   \u2571   \u2500\u2500 - 1 \n    \u2571     2     \n  \u2572\u2571     k      "
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, 0)\n# Takes forever",
     "language": "python",
     "outputs": [],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "source": "8"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 58
    },
    {
     "cell_type": "code",
     "input": "exp(-a**2*r**2/2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAVBAMAAAAOWFv7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDKr\n3c2gcmbwAAABAUlEQVQoFWWQsU4CQRRF72ZdYBYT0YpykWhP1g+YGPvVxhg6Ej+AwsZuEzpiseEH\nmFASbY2FMSbW9savsLUaZ2bfe8C6Bee8c4EQgK3necv/afRRp9UokCBvmwVTWVs7IciIeu2+J6Vr\nhOaKql35RpC5/iwwqQuBZ1pTEwKBx85nL+i8W3oSQuKX45MLa6vF+HHPgSMzzpZezfc+F0R5nrt/\nwFr7U1RvPqupjCzpsIcnnPozveQoXOAOD/jyPy3OqEaDM7JX3CA+HBp3JpraNc7JFO7JBK0pbvlo\n+S/deQ6uBprDmkVYlKId8yJOstRQpIX9ba6JQb/ZNrcaHenN1bA/ps4xpQOMXLQAAAAASUVORK5C\nYII=\n",
       "prompt_number": 59,
       "text": "\n   2  2\n -a \u22c5r \n \u2500\u2500\u2500\u2500\u2500\u2500\n   2   \n\u212f      "
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACwAAAAzBAMAAAAeKvd8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVN0y\nzasyv2LdAAABUklEQVQ4EWNgQAOLDNEEwFxOBXYHLOI8D1gbsAgzTGCfgC7MsgcokoguysBwgYGB\nawGm8AYGhi6eBnRxHks5lv//McxmesWIrhLE50vGUAkS5mVaAaLQAR9PBroQFj6jsbGxIQP/fxj4\ngEXNJIYSLKIM5xgiUIUhQc3JUIkiDAtqNgGQMKOiCUQWFtSrwNwwBieIMCiop8kd4FhwAshnK2BI\ngQoDgzqEdYL//59APn+gogNUmGsBywVeKNu/AcoABXXPg3NQ7nwHBk6IBCioDeXAzmBgYF3AIAER\nRiE5DYUcUASGCAcWf6g09R1/SO8AFkM5FXhQIx2ihucDww8sqhkOcH7BJszAnIBVWBhDVElqAwOH\nAjRa4bKsCswbGA5NkYELQBjSDMwGwKhHT9llDNwL0FQCuSw/GOZPwBTm+ckgzYUpzFDCco8Ji3Cz\n4DwVJGEAfiBeZy37B8AAAAAASUVORK5CYII=\n",
       "prompt_number": 60,
       "text": "\n   2 \n -k  \n \u2500\u2500\u2500\u2500\n    2\n 2\u22c5a \n\u212f    \n\u2500\u2500\u2500\u2500\u2500\n   2 \n  a  "
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAVBAMAAAAOWFv7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDKr\n3c2gcmbwAAABAUlEQVQoFWWQsU4CQRRF72ZdYBYT0YpykWhP1g+YGPvVxhg6Ej+AwsZuEzpiseEH\nmFASbY2FMSbW9savsLUaZ2bfe8C6Bee8c4EQgK3necv/afRRp9UokCBvmwVTWVs7IciIeu2+J6Vr\nhOaKql35RpC5/iwwqQuBZ1pTEwKBx85nL+i8W3oSQuKX45MLa6vF+HHPgSMzzpZezfc+F0R5nrt/\nwFr7U1RvPqupjCzpsIcnnPozveQoXOAOD/jyPy3OqEaDM7JX3CA+HBp3JpraNc7JFO7JBK0pbvlo\n+S/deQ6uBprDmkVYlKId8yJOstRQpIX9ba6JQb/ZNrcaHenN1bA/ps4xpQOMXLQAAAAASUVORK5C\nYII=\n",
       "prompt_number": 61,
       "text": "\n   2  2\n -a \u22c5r \n \u2500\u2500\u2500\u2500\u2500\u2500\n   2   \n\u212f      "
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "source": "9"
    },
    {
     "cell_type": "code",
     "input": "f = Function(\"f\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "input": "-r**2*f(r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAUBAMAAAAgmk0yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2VBBmq0Tvmd2J\nMs0OqBafAAABJ0lEQVQoFWNgQABe41oEBysrmOFmAFYJuOB9Bv4GOCcBzIKQcEFehngBGIcdwoqC\n8WH0GiCDdZkikEyBCLEYwKQgNNsCIM2UsBZIboCIMDyF0lCqmSGAgYEfZD5crxmKCl5TsQtQFewN\nUJl7cBXOhsYG8v//MzDwrq8CGsIPNMZY2BjMgKhhTfzMrgBh8gPNYYhnYAiTNyhiYOAMgIgycIRM\nYIUywSokGRhaPUGu4hZgEFICAmUGrgKoAgawCmkgbzlIhBtkIhhwJ0AZEBWRQN42kAjcFqTQBJsB\ndCnrR5AKIAMKPGEMiBl8FxhYJoCELsHFbWAs3vo3QCazAQPzAZBQGojABlg3QEXRQh1J7QkIm+MA\nkhgqkz0AzA9HFUXmsRqAeaYMDABQdy5hGjZx8QAAAABJRU5ErkJggg==\n",
       "prompt_number": 63,
       "text": "\n  2     \n-r \u22c5f(r)"
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "source": "10"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "input": "sin(2*pi*a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAnBAMAAABTf2O8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nY9xfzg2OOkWV/EYAAAAASUVORK5CYII=\n",
       "prompt_number": 66,
       "text": "\nsin(2\u22c5\u03c0\u22c5a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     r      "
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAdcAAADdCAMAAADetsghAAAAS1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 67,
       "text": "\n\u23a7                                                                 \u2502 2\u2502    \n\u23aa                                                               2 \u2502a \u2502    \n\u23aa                          0                             for 4\u22c5\u03c0 \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa                                                                 \u2502 2\u2502    \n\u23aa                                                                 \u2502k \u2502    \n\u23aa                                                                         \n\u23aa                                                              \u2502 2\u2502       \n\u23aa                                                              \u2502k \u2502       \n\u23aa                                                              \u2502\u2500\u2500\u2502       \n\u23aa                                                              \u2502 2\u2502       \n\u23aa                          1                                   \u2502a \u2502       \n\u23aa                \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                    for \u2500\u2500\u2500\u2500 < 1   \n\u23aa                        ______________                           2       \n\u23aa                       \u2571    2  2    2                         4\u22c5\u03c0        \n\u23aa                      \u2571  4\u22c5\u03c0 \u22c5a  - k                                     \n\u23aa                a\u22c5   \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                    \n\u23aa                    \u2571          2                                         \n\u23aa                  \u2572\u2571          a                                          \n\u23aa                                                                         \n\u23aa      \u23a7                                       \u2502 2\u2502                       \n\u23aa  ___ \u23aa                                     2 \u2502a \u2502                       \n\u23aa\u2572\u2571 \u03c0 \u22c5\u23aa             0                for 4\u22c5\u03c0 \u22c5\u2502\u2500\u2500\u2502 < 1                   \n\u23aa      \u23aa                                       \u2502 2\u2502                       \n\u23a8      \u23aa                                       \u2502k \u2502                       \n\u23aa      \u23aa                                                                  \n\u23aa      \u23aa                                    \u2502 2\u2502                          \n\u23aa      \u23aa                                    \u2502k \u2502                          \n\u23aa      \u23aa                                    \u2502\u2500\u2500\u2502                          \n\u23aa      \u23aa                                    \u2502 2\u2502                          \n\u23aa      \u23aa             k                      \u2502a \u2502                          \n\u23aa      \u23aa  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500      for \u2500\u2500\u2500\u2500 < 1                      \n\u23aa      \u23a8               ___________             2                          \n\u23aa      \u23aa              \u2571        2            4\u22c5\u03c0                           \n\u23aa      \u23aa   3/2       \u2571        k                                           \n\u23aa      \u23aa  \u03c0   \u22c5a\u22c5   \u2571   4 - \u2500\u2500\u2500\u2500\u2500                                         \n\u23aa      \u23aa           \u2571         2  2                                         \n\u23aa      \u23aa         \u2572\u2571         \u03c0 \u22c5a                                          \n\u23aa      \u23aa                                                                  \n\u23aa      \u23aa        \u239b         \u2502    2  2\u239e                                      \n\u23aa      \u23aa\u256d\u2500\u256e1, 1 \u239c1/2  1/2 \u2502 4\u22c5\u03c0 \u22c5a \u239f                                      \n\u23aa      \u23aa\u2502\u2576\u2510     \u239c         \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f      otherwise                       \n\u23aa      \u23aa\u2570\u2500\u256f2, 2 \u239c1/2   0  \u2502     2  \u239f                                      \n\u23aa      \u23aa        \u239d         \u2502    k   \u23a0                                      \n\u23aa      \u23a9                                                                  \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500      otherwise    \n\u23aa                          k                                              \n\u23a9                                                                         "
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "source": "10b"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "input": "cos(2*pi*a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAnBAMAAABqGNLnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nd4QqFqgzXAAAAABJRU5ErkJggg==\n",
       "prompt_number": 69,
       "text": "\ncos(2\u22c5\u03c0\u22c5a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     r      "
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAdcAAADdCAMAAADetsghAAAAS1BMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 70,
       "text": "\n\u23a7                                                                 \u2502 2\u2502    \n\u23aa                          1                                    2 \u2502a \u2502    \n\u23aa               \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                  for 4\u22c5\u03c0 \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa                       _______________                           \u2502 2\u2502    \n\u23aa                      \u2571      2  2                                \u2502k \u2502    \n\u23aa                     \u2571    4\u22c5\u03c0 \u22c5a                                         \n\u23aa               k\u22c5   \u2571   - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 1                                    \n\u23aa                   \u2571          2                                          \n\u23aa                 \u2572\u2571          k                                           \n\u23aa                                                                         \n\u23aa                                                              \u2502 2\u2502       \n\u23aa                                                              \u2502k \u2502       \n\u23aa                                                              \u2502\u2500\u2500\u2502       \n\u23aa                                                              \u2502 2\u2502       \n\u23aa                                                              \u2502a \u2502       \n\u23aa                          0                               for \u2500\u2500\u2500\u2500 < 1   \n\u23aa                                                                 2       \n\u23aa                                                              4\u22c5\u03c0        \n\u23aa                                                                         \n\u23aa      \u23a7                                       \u2502 2\u2502                       \n\u23aa  ___ \u23aa             1                       2 \u2502a \u2502                       \n\u23aa\u2572\u2571 \u03c0 \u22c5\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   for 4\u22c5\u03c0 \u22c5\u2502\u2500\u2500\u2502 < 1                   \n\u23aa      \u23aa            _______________            \u2502 2\u2502                       \n\u23a8      \u23aa           \u2571      2  2                 \u2502k \u2502                       \n\u23aa      \u23aa  ___     \u2571    4\u22c5\u03c0 \u22c5a                                             \n\u23aa      \u23aa\u2572\u2571 \u03c0 \u22c5   \u2571   - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + 1                                        \n\u23aa      \u23aa        \u2571          2                                              \n\u23aa      \u23aa      \u2572\u2571          k                                               \n\u23aa      \u23aa                                                                  \n\u23aa      \u23aa                                    \u2502 2\u2502                          \n\u23aa      \u23aa                                    \u2502k \u2502                          \n\u23aa      \u23a8                                    \u2502\u2500\u2500\u2502                          \n\u23aa      \u23aa                                    \u2502 2\u2502                          \n\u23aa      \u23aa                                    \u2502a \u2502                          \n\u23aa      \u23aa             0                  for \u2500\u2500\u2500\u2500 < 1                      \n\u23aa      \u23aa                                       2                          \n\u23aa      \u23aa                                    4\u22c5\u03c0                           \n\u23aa      \u23aa                                                                  \n\u23aa      \u23aa        \u239b         \u2502    2  2\u239e                                      \n\u23aa      \u23aa\u256d\u2500\u256e1, 1 \u239c1/2  1/2 \u2502 4\u22c5\u03c0 \u22c5a \u239f                                      \n\u23aa      \u23aa\u2502\u2576\u2510     \u239c         \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f      otherwise                       \n\u23aa      \u23aa\u2570\u2500\u256f2, 2 \u239c 0   1/2 \u2502     2  \u239f                                      \n\u23aa      \u23aa        \u239d         \u2502    k   \u23a0                                      \n\u23aa      \u23a9                                                                  \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500      otherwise    \n\u23aa                          k                                              \n\u23a9                                                                         "
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "markdown",
     "source": "11"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "input": "(a**2+r**2)**(-Rational(3,2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAeBAMAAACxjX91AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9qFsXdNhWP0XFjrZFZ2ET6ydVkxt/1o7AAAAAElFTkSuQmCC\n",
       "prompt_number": 72,
       "text": "\n     1      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         3/2\n\u239b 2    2\u239e   \n\u239da  + r \u23a0   "
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAnBAMAAABUEX/sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 73,
       "text": "\n-sinh(a\u22c5k) + cosh(a\u22c5k)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          a           "
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 74
    },
    {
     "cell_type": "markdown",
     "source": "12"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "input": "2*a**2/(a**2+r**2)**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAvBAMAAACRacBDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 76,
       "text": "\n      2   \n   2\u22c5a    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2\n\u239b 2    2\u239e \n\u239da  + r \u23a0 "
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAuCAMAAACVtZLwAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 77,
       "text": "\n        \u239b        \u2502  2  2\u239e\n\u256d\u2500\u256e2, 1 \u239c 0      \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c        \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d1, 0  0 \u2502   4  \u23a0"
      }
     ],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "input": "# Should be a besselk of order 1",
     "language": "python",
     "outputs": [],
     "prompt_number": 78
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAA6BAMAAAC0SOHgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 79,
       "text": "\n     2      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           2\n   \u239b     2\u239e \n 2 \u239c    r \u239f \na \u22c5\u239c1 + \u2500\u2500\u239f \n   \u239c     2\u239f \n   \u239d    a \u23a0 "
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "markdown",
     "source": "13"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "input": "4*a**4/(a**2+r**2)**3",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAvBAMAAACRacBDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXoJ3dMBDfdYnVz7rH8sxkhipih3XAAAAAElFTkSuQmCC\n",
       "prompt_number": 81,
       "text": "\n      4   \n   4\u22c5a    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         3\n\u239b 2    2\u239e \n\u239da  + r \u23a0 "
      }
     ],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAuCAMAAACVtZLwAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nF7d68l8/d8diuymY6yzKwvQvDv4BklAjhOxhooYAAAAASUVORK5CYII=\n",
       "prompt_number": 82,
       "text": "\n        \u239b        \u2502  2  2\u239e\n\u256d\u2500\u256e2, 1 \u239c 0      \u2502 a \u22c5k \u239f\n\u2502\u2576\u2510     \u239c        \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d2, 0  0 \u2502   4  \u23a0"
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "input": "# Should be a besselk of order 2",
     "language": "python",
     "outputs": [],
     "prompt_number": 83
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAA6BAMAAAC0SOHgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 84,
       "text": "\n     4      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           3\n   \u239b     2\u239e \n 2 \u239c    r \u239f \na \u22c5\u239c1 + \u2500\u2500\u239f \n   \u239c     2\u239f \n   \u239d    a \u23a0 "
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "markdown",
     "source": "14"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 85
    },
    {
     "cell_type": "code",
     "input": "exp(-a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAANBAMAAADVpBoRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAcklEQVQIHWNggAMmpZBJUevgXCDDiPPDglt8EBFGY2NjQ5af3Ac4C5CVcF1gSuAK\nQBbheMDbyKwAEmFUNIFIGApZsTqAmGEMThARGMlWwJACY0No/kBFB1QR/wZUPgPDfAcGTlQx1gUM\nEqginIZCDigiAB0NEUwYAF3HAAAAAElFTkSuQmCC\n",
       "prompt_number": 86,
       "text": "\n -a\u22c5r\n\u212f    "
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAA5BAMAAAAo4HPAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 87,
       "text": "\n      a       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           3/2\n   \u239b 2    \u239e   \n 3 \u239ca     \u239f   \nk \u22c5\u239c\u2500\u2500 + 1\u239f   \n   \u239c 2    \u239f   \n   \u239dk     \u23a0   "
      }
     ],
     "prompt_number": 87
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAASBAMAAADf6JPtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2ZlTNRImrEN0y\n75nBkDP4AAACB0lEQVQ4EZWTP0gbYRjGn2tOktzFJAjF8eQ62iFygRYHOUksiLUNxApCKw6C2Ha4\npaVQqIKLYzoVXHJQOojgZet6SFvolm4dOghuDsU/pUIJ6vfm+9LvLl4u+kJenvf9fs9D+O4OiClV\nntlS9lL9acUX3vcyY1fKLnUdetZtm/Za3KtWZMZAQeqwug4973KPCNaDCc/58DW460t3wyK4GtwX\n+TAX3HEdQ3fDAv0Q3H/jQ+/gKJo86pMHmVH/9XH5AGhRA47Yz5pqoDpmFZCjDSCCFy0b1ZKNqbml\nOJocb5B2PV87heHinBo0FnyrkG1ok6fpEQyyDSsenFhSfqX2sZlcx1YcTY63ThI5H3/+NyhNYAbZ\n/eSnpgboeaJEsFfRDo08jOHPcLjlKj1ksroDpC/uUmYLOYc3KA3gJ/QKsjWWqPtA1jRfmCab1tjw\nGMjZ9RNhiaCZicq9Xc+Hg7UmlDN4LnSbnYeuohM8riwci78RSVPwQ6QK4WD28DL/MDMAj24h9PC8\nChTDh/HFx4YIjqQpeBoJhwWf0VW0G/AS+KGsJvCMzveode440cBkqobfag334mhylIrFTH3l+9+n\n9eV2YysLeFfeHsN9Oi9T6wRjwXLwaMJWrZIdR3PPlb4T3IhP+lVwF9JRdAiQQ8qRmr2xfeom9JbM\n2payl7oB/VFmjEvZS4XpS5tGryjOT0kMAAAAAElFTkSuQmCC\n",
       "prompt_number": 88,
       "text": "-sinh(a\u22c5r) + cosh(a\u22c5r)"
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "markdown",
     "source": "15"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 89
    },
    {
     "cell_type": "code",
     "input": "r**2*exp(-pi*r**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA60lEQVQoFWNgwAmEluKUYmBgUeA0wC3Nm8DegFuWwYHTASJ7Se0CFmVHIWIsCrxT\nMGUZBSBivB8YfmDKNvI2QAQvsHxGyPLMnz+PRV0w+f9/B6gg1wGEbC8TEwMjt0ACQiQKwoxatWpV\nAMNFHgYGT4QcA4cCCxJviRcDgzgS/5JzLILH8sCOgWERhM+oZKTE/P//BwaGoChIyDEqcDMwFENk\nXe0V0sEs9gPMChAhJLJFksEBzD3JwAPRiyTJwLAMyvseFIwiDuFshVDMWEILGI3QUALRDhiaGSdA\nhQoZWBswZJkOQIW8QpUxJAEVOyvPUN4XXQAAAABJRU5ErkJggg==\n",
       "prompt_number": 90,
       "text": "\n        2\n 2  -\u03c0\u22c5r \nr \u22c5\u212f     "
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAyBAMAAACAOwXCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 91,
       "text": "\n                2\n              -k \n              \u2500\u2500\u2500\n\u239b   2      \u239e  4\u22c5\u03c0\n\u239d- k  + 4\u22c5\u03c0\u23a0\u22c5\u212f   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          3      \n       8\u22c5\u03c0       "
      }
     ],
     "prompt_number": 91
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA60lEQVQoFWNgwAmEluKUYmBgUeA0wC3Nm8DegFuWwYHTASJ7Se0CFmVHIWIsCrxT\nMGUZBSBivB8YfmDKNvI2QAQvsHxGyPLMnz+PRV0w+f9/B6gg1wGEbC8TEwMjt0ACQiQKwoxatWpV\nAMNFHgYGT4QcA4cCCxJviRcDgzgS/5JzLILH8sCOgWERhM+oZKTE/P//BwaGoChIyDEqcDMwFENk\nXe0V0sEs9gPMChAhJLJFksEBzD3JwAPRiyTJwLAMyvseFIwiDuFshVDMWEILGI3QUALRDhiaGSdA\nhQoZWBswZJkOQIW8QpUxJAEVOyvPUN4XXQAAAABJRU5ErkJggg==\n",
       "prompt_number": 92,
       "text": "\n        2\n 2  -\u03c0\u22c5r \nr \u22c5\u212f     "
      }
     ],
     "prompt_number": 92
    },
    {
     "cell_type": "markdown",
     "source": "Arbitrary order"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 93
    },
    {
     "cell_type": "code",
     "input": "exp(-a**2*r**2/2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAVBAMAAAAOWFv7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDKr\n3c2gcmbwAAABAUlEQVQoFWWQsU4CQRRF72ZdYBYT0YpykWhP1g+YGPvVxhg6Ej+AwsZuEzpiseEH\nmFASbY2FMSbW9savsLUaZ2bfe8C6Bee8c4EQgK3necv/afRRp9UokCBvmwVTWVs7IciIeu2+J6Vr\nhOaKql35RpC5/iwwqQuBZ1pTEwKBx85nL+i8W3oSQuKX45MLa6vF+HHPgSMzzpZezfc+F0R5nrt/\nwFr7U1RvPqupjCzpsIcnnPozveQoXOAOD/jyPy3OqEaDM7JX3CA+HBp3JpraNc7JFO7JBK0pbvlo\n+S/deQ6uBprDmkVYlKId8yJOstRQpIX9ba6JQb/ZNrcaHenN1bA/ps4xpQOMXLQAAAAASUVORK5C\nYII=\n",
       "prompt_number": 94,
       "text": "\n   2  2\n -a \u22c5r \n \u2500\u2500\u2500\u2500\u2500\u2500\n   2   \n\u212f      "
      }
     ],
     "prompt_number": 94
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAqwAAAA6BAMAAACHTBZ0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 95,
       "text": "\n            \u239b                                                                \u239b\n            \u239c                                                       2        \u239c\n            \u239c                                                      k \u22c5besseli\u239c\n            \u239c            \u239b         2 \u239e             \u239b         2 \u239e             \u239c\n  ___   ___ \u239c 2          \u239c\u03bd   1   k  \u239f    2        \u239c\u03bd   1   k  \u239f             \u239d\n\u2572\u2571 2 \u22c5\u2572\u2571 \u03c0 \u22c5\u239ca \u22c5\u03bd\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u2500\u239f + a \u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u2500\u239f - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            \u239c            \u239c2   2     2\u239f             \u239c2   2     2\u239f              \n            \u239d            \u239d       4\u22c5a \u23a0             \u239d       4\u22c5a \u23a0              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                             3                \n                                                            a \u22c5k              \n\n         2 \u239e             \u239b         2 \u239e\u239e    2 \n\u03bd   1   k  \u239f    2        \u239c\u03bd   3   k  \u239f\u239f  -k  \n\u2500 + \u2500, \u2500\u2500\u2500\u2500\u239f   k \u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u2500\u239f\u239f  \u2500\u2500\u2500\u2500\n2   2     2\u239f             \u239c2   2     2\u239f\u239f     2\n       4\u22c5a \u23a0             \u239d       4\u22c5a \u23a0\u239f  4\u22c5a \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u212f    \n4                         4           \u239f      \n                                      \u23a0      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                             \n                                             "
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 96
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "s = Symbol(\"s\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 97
    },
    {
     "cell_type": "code",
     "input": "r**s",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAA4AAAANBAMAAABr8kJMAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVHYiEGaru0Tvmd2Jzd8e\nNcIAAABGSURBVAgdY2AAAVFBMMVSYAah5yqAaQbGS2DamiGQgVHJSMlJNIDB1U4hHSTYIsngAJZc\nBiYZGLZCaJbHEJpxAoRmOgCmAdfvCYuhScOJAAAAAElFTkSuQmCC\n",
       "prompt_number": 98,
       "text": "\n s\nr "
      }
     ],
     "prompt_number": 98
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM8AAAAxBAMAAACPN+AFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 99,
       "text": "\n s + 1  -s - 2  \u239b\u03bd   s    \u239e\n2     \u22c5k      \u22c5\u0393\u239c\u2500 + \u2500 + 1\u239f\n                \u239d2   2    \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n           \u239b\u03bd   s\u239e         \n          \u0393\u239c\u2500 - \u2500\u239f         \n           \u239d2   2\u23a0         "
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "s = Symbol(\"s\")\nh = Symbol(\"h\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 100
    },
    {
     "cell_type": "markdown",
     "source": "Can not express $r^{\\nu-2s}\\Gamma\\left(s,r^2 h\\right)$"
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "markdown",
     "source": "Can not express $e^{-r^2}r^\\nu U\\left(a,b,r^2\\right)$"
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 101
    },
    {
     "cell_type": "code",
     "input": "4*a**4/(a**2+r**2)**3",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAvBAMAAACRacBDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXoJ3dMBDfdYnVz7rH8sxkhipih3XAAAAAElFTkSuQmCC\n",
       "prompt_number": 102,
       "text": "\n      4   \n   4\u22c5a    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         3\n\u239b 2    2\u239e \n\u239da  + r \u23a0 "
      }
     ],
     "prompt_number": 102
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABY4AAABJCAMAAACtmUJ0AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n3/RBUmdP21x0s+yEQBAQBASBfRH4P4x7JD5/KFIYAAAAAElFTkSuQmCC\n",
       "prompt_number": 103,
       "text": "\n                    \u239b      3        \u2502  2  2\u239e             \u239b           \u239b  2  2  \n 4  4  \u239b\u03bd    \u239e  \u250c\u2500  \u239c               \u2502 a \u22c5k \u239f    -\u03bd  \u03bd  \u03bd \u239c \u03bd  -\u03bd  -\u03bd \u239c a \u22c5k   \na \u22c5k \u22c5\u0393\u239c\u2500 - 2\u239f\u22c5 \u251c\u2500  \u239c\u03bd        \u03bd     \u2502 \u2500\u2500\u2500\u2500\u2500\u239f   2  \u22c5a \u22c5k \u22c5\u239c2 \u22c5a  \u22c5k  \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       \u239d2    \u23a0 1\u2575 2 \u239c\u2500 + 3, - \u2500 + 3 \u2502   4  \u239f             \u239c           \u239c 2      \n                    \u239d2        2     \u2502      \u23a0             \u239d           \u239d\u03bd  - 2\u22c5\u03bd\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    \u239b\u03bd    \u239e                                                   \n                 8\u22c5\u0393\u239c\u2500 + 3\u239f                                                   \n                    \u239d2    \u23a0                                                   \n\n    \u239e                                                                         \n    \u239f                             \u03bd - 1  -\u03bd + 1  -\u03bd + 1 \u239b    4\u22c5\u03bd      4\u22c5\u03bd - 4 \n + 1\u239f\u22c5besseli(\u03bd, a\u22c5k)\u22c5\u0393(\u03bd + 1) + 2     \u22c5a      \u22c5k      \u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    \u239f                                                   \u239c   2          2      \n    \u23a0                                                   \u239d  \u03bd  - 2\u22c5\u03bd   \u03bd  - 2\u22c5\u03bd\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                             \u0393(\u03bd + 1)                         \n                                                                              \n                                                                              \n\n                              \u239e                    \n\u239e                             \u239f  \u239b  \u03bd    \u239e  \u239b\u03bd    \u239e\n\u239f\u22c5besseli(\u03bd + 1, a\u22c5k)\u22c5\u0393(\u03bd + 1)\u239f\u22c5\u0393\u239c- \u2500 + 2\u239f\u22c5\u0393\u239c\u2500 + 1\u239f\n\u239f                             \u239f  \u239d  2    \u23a0  \u239d2    \u23a0\n\u23a0                             \u23a0                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                   \n                                                   \n                                                   "
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "code",
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABY4AAABJCAMAAACtmUJ0AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n3/RBUmdP21x0s+yEQBAQBASBfRH4P4x7JD5/KFIYAAAAAElFTkSuQmCC\n",
       "prompt_number": 104,
       "text": "\n                    \u239b      3        \u2502  2  2\u239e             \u239b           \u239b  2  2  \n 4  4  \u239b\u03bd    \u239e  \u250c\u2500  \u239c               \u2502 a \u22c5k \u239f    -\u03bd  \u03bd  \u03bd \u239c \u03bd  -\u03bd  -\u03bd \u239c a \u22c5k   \na \u22c5k \u22c5\u0393\u239c\u2500 - 2\u239f\u22c5 \u251c\u2500  \u239c\u03bd        \u03bd     \u2502 \u2500\u2500\u2500\u2500\u2500\u239f   2  \u22c5a \u22c5k \u22c5\u239c2 \u22c5a  \u22c5k  \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       \u239d2    \u23a0 1\u2575 2 \u239c\u2500 + 3, - \u2500 + 3 \u2502   4  \u239f             \u239c           \u239c 2      \n                    \u239d2        2     \u2502      \u23a0             \u239d           \u239d\u03bd  - 2\u22c5\u03bd\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    \u239b\u03bd    \u239e                                                   \n                 8\u22c5\u0393\u239c\u2500 + 3\u239f                                                   \n                    \u239d2    \u23a0                                                   \n\n    \u239e                                                                         \n    \u239f                             \u03bd - 1  -\u03bd + 1  -\u03bd + 1 \u239b    4\u22c5\u03bd      4\u22c5\u03bd - 4 \n + 1\u239f\u22c5besseli(\u03bd, a\u22c5k)\u22c5\u0393(\u03bd + 1) + 2     \u22c5a      \u22c5k      \u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    \u239f                                                   \u239c   2          2      \n    \u23a0                                                   \u239d  \u03bd  - 2\u22c5\u03bd   \u03bd  - 2\u22c5\u03bd\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                             \u0393(\u03bd + 1)                         \n                                                                              \n                                                                              \n\n                              \u239e                    \n\u239e                             \u239f  \u239b  \u03bd    \u239e  \u239b\u03bd    \u239e\n\u239f\u22c5besseli(\u03bd + 1, a\u22c5k)\u22c5\u0393(\u03bd + 1)\u239f\u22c5\u0393\u239c- \u2500 + 2\u239f\u22c5\u0393\u239c\u2500 + 1\u239f\n\u239f                             \u239f  \u239d  2    \u23a0  \u239d2    \u23a0\n\u23a0                             \u23a0                    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                   \n                                                   \n                                                   "
      }
     ],
     "prompt_number": 104
    },
    {
     "cell_type": "markdown",
     "source": "6"
    },
    {
     "cell_type": "code",
     "input": "1/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAlBAMAAACJ5mVtAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEKtEMol2VCK73Zlm72LS\nT58AAABKSURBVAgdY2BgVGZgYGAJew4kGRjIITkiX20GayaSeAcCRKpFV8ae1pbGwCDTl6DJwGBY\nyyAAlN8EVrMGRDK+AJHsF0AkWwCQAADSdRbnwMlO3gAAAABJRU5ErkJggg==\n",
       "prompt_number": 105,
       "text": "\n1\n\u2500\nr"
      }
     ],
     "prompt_number": 105
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAlBAMAAABmJA5TAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAAAaElEQVQYGWNgYFRmAAKWsO8gioGBIooj8tdmiDGkkf9B4AN+PTUfwfLsF8AUqwKY\n4nkApuQbwJQtQ+kBICOITXwBkDr6AEgwMP5MAVHMF+QdgBSrAs+DAgYGngdchg8YGLga2LcxMAAA\nX08hl99B2lwAAAAASUVORK5CYII=\n",
       "prompt_number": 106,
       "text": "\n1\n\u2500\nk"
      }
     ],
     "prompt_number": 106
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAlBAMAAACJ5mVtAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEKtEMol2VCK73Zlm72LS\nT58AAABKSURBVAgdY2BgVGZgYGAJew4kGRjIITkiX20GayaSeAcCRKpFV8ae1pbGwCDTl6DJwGBY\nyyAAlN8EVrMGRDK+AJHsF0AkWwCQAADSdRbnwMlO3gAAAABJRU5ErkJggg==\n",
       "prompt_number": 107,
       "text": "\n1\n\u2500\nr"
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "markdown",
     "source": "7"
    },
    {
     "cell_type": "code",
     "input": "r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAIBAMAAAA2IaO4AAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAVHYiEGaru0Tvmd2JcbBxRwAA\nACtJREFUCB1jYFQyUmJwtVVIZ2iRZHBgYFjGAARbgZjlMpBgnAAkmA4wMAAAeHEF42gIMlAAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 108,
       "text": "r"
      }
     ],
     "prompt_number": 108
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 109
    },
    {
     "cell_type": "markdown",
     "source": "7b"
    },
    {
     "cell_type": "code",
     "input": "m = Symbol(\"m\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 110
    },
    {
     "cell_type": "code",
     "input": "1/r**m",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAANBAMAAADVpBoRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAb0lEQVQIHWNggAOn5KC1QQpwLpAhsInhI8sEqEjUqlWrAlgmME/gKEBWw1jAuoAr\nAFmES4FNgNOJgYFRyUgJIs5mwOfApcvA4GqvkI6skoGhRZLBAVWEYRkan4FhK7oIy2d0EUaYK+ES\nTAfgTAgDADWBEV+IwoMIAAAAAElFTkSuQmCC\n",
       "prompt_number": 111,
       "text": "\n -m\nr  "
      }
     ],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPAAAAAxBAMAAAAWzzmwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 112,
       "text": "\n -m + 1  m - 2  \u239b  m   \u03bd    \u239e\n2      \u22c5k     \u22c5\u0393\u239c- \u2500 + \u2500 + 1\u239f\n                \u239d  2   2    \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n            \u239bm   \u03bd\u239e          \n           \u0393\u239c\u2500 + \u2500\u239f          \n            \u239d2   2\u23a0          "
      }
     ],
     "prompt_number": 112
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAANBAMAAADVpBoRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAb0lEQVQIHWNggAOn5KC1QQpwLpAhsInhI8sEqEjUqlWrAlgmME/gKEBWw1jAuoAr\nAFmES4FNgNOJgYFRyUgJIs5mwOfApcvA4GqvkI6skoGhRZLBAVWEYRkan4FhK7oIy2d0EUaYK+ES\nTAfgTAgDADWBEV+IwoMIAAAAAElFTkSuQmCC\n",
       "prompt_number": 113,
       "text": "\n -m\nr  "
      }
     ],
     "prompt_number": 113
    },
    {
     "cell_type": "markdown",
     "source": "8"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 114
    },
    {
     "cell_type": "code",
     "input": "1/sqrt(r**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAqBAMAAADi/wAlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73ZnN\n72bDLMX7AAABX0lEQVQ4Ee2UP0vDQBjGn2tiL2kQpYs6taNjd5cg2FXBLxAESzed3MQquFe3gmDX\n4tJBBDfxE+ggON4gKBWkUvHPFN+73KUhgdBBEMQbnvd5f89LuLuQAOlVrqVJtl/Z2s7CDJn5H/qx\nKzg8PZvgvWRewZ8H4QTrly5hNW9rZk9HxuRUu6/C+oafHkogtyVDFti3qaEkWlaZPcQoNZREi1Hm\ns89oevwVGwQ4Vf0Ab00ZLnRPRSOg4FMn13NUuIiqVI2AA2oaj11YAZMcXJAAO81GECE5cA64gddF\nff5BhuCCBOzpvdBWyKL/EWsDA3h9JwyHMgUXJLDmbphG7hWmfOAOJXOky85Jr3NMU16VRC3rDQt0\nvhEqswaBC2VLUZG+h3XAfsWAq0QKFyRApaWKlL192hLunetijLhQdjcGmL6oUbP0Um7GjAtlN2OA\n4tfYa2eZM8QJ+4htjvFzMhV9A217jHtwiRTvAAAAAElFTkSuQmCC\n",
       "prompt_number": 115,
       "text": "\n     1      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         1/2\n\u239b 2    2\u239e   \n\u239da  + r \u23a0   "
      }
     ],
     "prompt_number": 115
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAEzIAAAA+BAMAAACsx/czAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7f0l/HGmCG90FFW0qAwoA8rA5xwD/wdsww7Jed+3HQAAAABJRU5ErkJggg==\n",
       "prompt_number": 116,
       "text": "\n  \u239b                                                                           \n  \u239c 2  2        \u239b  \u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e             2  2        \u239b  \n\u03c0\u22c5\u239ca \u22c5k \u22c5besseli\u239c- \u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) + a \u22c5k \u22c5besseli\u239c- \n  \u239d             \u239d  2   2   2 \u23a0        \u239d2   2   2 \u23a0                         \u239d  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n\u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e               2  2        \u239b\u03bd   1  a\u22c5k\u239e        \n\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) - 2\u22c5a \u22c5k \u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\n2   2   2 \u23a0        \u239d2   2   2 \u23a0                           \u239d2   2   2 \u23a0        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                      2                                      2\n\u239b\u03bd   1  a\u22c5k\u239e                 2         \u239b\u03bd   1  a\u22c5k\u239e                 2         \n\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) - a\u22c5k\u22c5\u03bd \u22c5besseli \u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) + a\u22c5k\u22c5\u03bd \u22c5besseli \n\u239d2   2   2 \u23a0                           \u239d2   2   2 \u23a0                           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                     2                                       2\n\u239b\u03bd   1  a\u22c5k\u239e                          \u239b\u03bd   1  a\u22c5k\u239e                            \n\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) + a\u22c5k\u22c5\u03bd\u22c5besseli \u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2) + 2\u22c5a\u22c5k\u22c5\u03bd\u22c5besseli \n\u239d2   2   2 \u23a0                          \u239d2   2   2 \u23a0                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                     2                                   2    \n\u239b\u03bd   1  a\u22c5k\u239e                          \u239b\u03bd   1  a\u22c5k\u239e                        \u239b\u03bd  \n\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) - a\u22c5k\u22c5\u03bd\u22c5besseli \u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2) + a\u22c5k\u22c5besseli \u239c\u2500 -\n\u239d2   2   2 \u23a0                          \u239d2   2   2 \u23a0                        \u239d2  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                               2                                   2          \n 1  a\u22c5k\u239e                        \u239b\u03bd   1  a\u22c5k\u239e                        \u239b\u03bd   1  a\u22c5\n \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) - a\u22c5k\u22c5besseli \u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2) + a\u22c5k\u22c5besseli \u239c\u2500 + \u2500, \u2500\u2500\n 2   2 \u23a0                        \u239d2   2   2 \u23a0                        \u239d2   2   2\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                          2    \u239b\u03c0\u22c5\u03bd\u239e                                          \n                     4\u22c5a\u22c5k \u22c5cos\u239c\u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1)                                 \n                               \u239d 2 \u23a0                                          \n\n                         2                                                    \nk\u239e                        \u239b\u03bd   1  a\u22c5k\u239e               3        \u239b\u03bd   1  a\u22c5k\u239e    \n\u2500\u239f\u22c5\u0393(\u03bd + 1) - a\u22c5k\u22c5besseli \u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2) + 2\u22c5\u03bd \u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5bes\n \u23a0                        \u239d2   2   2 \u23a0                        \u239d2   2   2 \u23a0    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n    \u239b\u03bd   1  a\u22c5k\u239e               2        \u239b\u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e      \nseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) + 2\u22c5\u03bd \u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd +\n    \u239d2   2   2 \u23a0                        \u239d2   2   2 \u23a0        \u239d2   2   2 \u23a0      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n         2        \u239b\u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e                       \u239b\u03bd   \n 1) - 2\u22c5\u03bd \u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2) - 2\u22c5\u03bd\u22c5besseli\u239c\u2500 - \n                  \u239d2   2   2 \u23a0        \u239d2   2   2 \u23a0                       \u239d2   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                              \n1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e                     \u239b\u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5\n\u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 1) - 2\u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\n2   2 \u23a0        \u239d2   2   2 \u23a0                     \u239d2   2   2 \u23a0        \u239d2   2   2\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n\n                                                                \u239e\nk\u239e                     \u239b\u03bd   1  a\u22c5k\u239e        \u239b\u03bd   1  a\u22c5k\u239e         \u239f\n\u2500\u239f\u22c5\u0393(\u03bd + 1) + 2\u22c5besseli\u239c\u2500 - \u2500, \u2500\u2500\u2500\u239f\u22c5besseli\u239c\u2500 + \u2500, \u2500\u2500\u2500\u239f\u22c5\u0393(\u03bd + 2)\u239f\n \u23a0                     \u239d2   2   2 \u23a0        \u239d2   2   2 \u23a0         \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                 \n                                                                 \n                                                                 "
      }
     ],
     "prompt_number": 116
    },
    {
     "cell_type": "markdown",
     "source": "9"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 117
    },
    {
     "cell_type": "code",
     "input": "1/(r**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAmBAMAAACMti74AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA/0lEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IFyL7OkqWAY\n4pbkABUTY/AVQJNmTGC5CBXyZ+A3QJNl+cDwGSrEwiDfgCbL4MD4Ay60FcJyhwswMHAHwDg8GyCs\nGpgAkD4NZ5sxQFyFJMucwAiUZsupYmBJ7HEAq4TIsqe1pTG4SRwBCk1j2MXQ//8/WJIBIivTn6DJ\n8f//BwYGngUMuyEyIBIia1gLtYiBN4BhOVSWe+bMVzNnKgB5m6AiDP0GjB9hbJhehjUwEX8BngUF\nMA7UZEQ48Db0KSSgybJfgAkw7zmo1gDjQPWyBcAFUBjmKDwGAOi7VkS1Y1DSAAAAAElFTkSuQmCC\n",
       "prompt_number": 118,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2\na  + r "
      }
     ],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAjwAAABJCAMAAAAOoQzzAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nKvQPaYQ5WXIkufq/C/sfS9yKYfShL5AAAAAASUVORK5CYII=\n",
       "prompt_number": 119,
       "text": "\n \u239b                     \u239b      1        \u2502  2  2\u239e                             \u239e\n \u239c     \u239b\u03c0\u22c5\u03bd\u239e  \u239b\u03bd\u239e  \u250c\u2500  \u239c               \u2502 a \u22c5k \u239f                      \u239b\u03bd    \u239e\u239f\n-\u239c- sin\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500\u239f\u22c5 \u251c\u2500  \u239c\u03bd        \u03bd     \u2502 \u2500\u2500\u2500\u2500\u2500\u239f + \u03c0\u22c5besseli(\u03bd, a\u22c5k)\u22c5\u0393\u239c\u2500 + 1\u239f\u239f\n \u239c     \u239d 2 \u23a0  \u239d2\u23a0 1\u2575 2 \u239c\u2500 + 1, - \u2500 + 1 \u2502   4  \u239f                      \u239d2    \u23a0\u239f\n \u239d                     \u239d2        2     \u2502      \u23a0                             \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  \u239b\u03c0\u22c5\u03bd\u239e  \u239b\u03bd    \u239e                             \n                             2\u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                             \n                                  \u239d 2 \u23a0  \u239d2    \u23a0                             "
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "code",
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAjwAAABJCAMAAAAOoQzzAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nKvQPaYQ5WXIkufq/C/sfS9yKYfShL5AAAAAASUVORK5CYII=\n",
       "prompt_number": 120,
       "text": "\n \u239b                     \u239b      1        \u2502  2  2\u239e                             \u239e\n \u239c     \u239b\u03c0\u22c5\u03bd\u239e  \u239b\u03bd\u239e  \u250c\u2500  \u239c               \u2502 a \u22c5k \u239f                      \u239b\u03bd    \u239e\u239f\n-\u239c- sin\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500\u239f\u22c5 \u251c\u2500  \u239c\u03bd        \u03bd     \u2502 \u2500\u2500\u2500\u2500\u2500\u239f + \u03c0\u22c5besseli(\u03bd, a\u22c5k)\u22c5\u0393\u239c\u2500 + 1\u239f\u239f\n \u239c     \u239d 2 \u23a0  \u239d2\u23a0 1\u2575 2 \u239c\u2500 + 1, - \u2500 + 1 \u2502   4  \u239f                      \u239d2    \u23a0\u239f\n \u239d                     \u239d2        2     \u2502      \u23a0                             \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  \u239b\u03c0\u22c5\u03bd\u239e  \u239b\u03bd    \u239e                             \n                             2\u22c5sin\u239c\u2500\u2500\u2500\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                             \n                                  \u239d 2 \u23a0  \u239d2    \u23a0                             "
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "markdown",
     "source": "Bessel functions"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "input": "besselj(nu-1, a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAnBAMAAACxo3jFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
       "prompt_number": 122,
       "text": "\nbesselj(\u03bd - 1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         r         "
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAABBCAMAAAD2WPjiAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 123,
       "text": "\n        \u239b 1/2     1/2   \u2502  2\u239e\n\u256d\u2500\u256e1, 1 \u239c               \u2502 a \u239f\n\u2502\u2576\u2510     \u239c\u03bd   1    \u03bd   1 \u2502 \u2500\u2500\u239f\n\u2570\u2500\u256f2, 2 \u239c\u2500 - \u2500  - \u2500 + \u2500 \u2502  2\u239f\n        \u239d2   2    2   2 \u2502 k \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n              k              "
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAnBAMAAACxo3jFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
       "prompt_number": 124,
       "text": "\nbesselj(\u03bd - 1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         r         "
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 125
    },
    {
     "cell_type": "code",
     "input": "besselj(nu, a*r)/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAnBAMAAABZI83uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+3OGqwgAAAAASUVORK5CYII=\n",
       "prompt_number": 126,
       "text": "\nbesselj(\u03bd, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2      \n       r       "
      }
     ],
     "prompt_number": 126
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAuCAMAAACClwI5AAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAEKtEMol2\nVCK73Zlm781Aod7tH+cAAATcSURBVFgJzVnZAqYqCNZKy7Y5vf/LjqgoCq3nv5huImX5VAQ0pf6R\nRw8NkMHarm1rWF58Tv0L5lvWwdQs2io1bNo36h/YmXbQ9LOnBTscXvURcC4rsaK7D9j1b7GqFqzy\nLqCPKcB0I6IdjD3yBzbev/dK5tV4RWYG1kNYkmvoOYIOqD6ANR0Zz6vxnjALYNcFbfQzUt41qlkq\n7edUdH3S/0qFxMzB9n6L4bZwnk6PJIx98ts1e/fdeCV7Gay16wgTOVg9rRi7hgNhv7ME6IlsGoxk\nP3Xxl8SMYGEOFw92OuDJsmV2JOHMJhGOemxgeKVCYk5gV5jUJTtrNt5n3JIwsLmyCQkJIQWXJys7\nU5EZKCExJ7Ab+Ncs7KENg60krNRoNgRLSDBqN2o60LIKxhYbJOY//0HfBGlAFwctCpY90ZIwdGWw\nFanUzrzgndtL9uLMhs3gXYFPbY8jkIQvwGqBX2hKMyG8JGZ0g0lNfi5KpMriQWgy7pgXMeGezOyI\ng8yKXobqC7D9sq66s+h/xMLcRkvSB+QJWNO67MV4G43+84Q5zSznx5bOIVXe2nSLMdME4zgB69DX\ni9T/p27B2hy8sjE7B4cwG7h4ANuHMEVwq5nvryz/mbgFO6YSrFhwqfCD6NHbo/PBbfHTn8jEF2Jh\nkfkNdQt2iMVtsbZk9GTyMBpntoO15K7vxC3YqdmW/SGEDB5GGqnvAKnkLVjdzJETYpIac72TdLdD\npCa/07dgVTOTR9zmdtlLkcPNM+fhLB9aYrq9EjyMCpVYKscO9NSDlz1FTQu2qnEK2yXFZeLMFjRI\nFTUeLH22BHYIXkkO7oSEYoNm7qbGoepOaUnmtRssKTWFoFAO7oqQHkDr6TQEn+JrOrjMA7BNENrD\n8vcdgCYHd0KC0WZbcsMNMuGTyzCw1Wp6Fbx8st1i7ajCGMjBnZAAtg5w3LCArmniMgh23GNIqlfT\ni4dSV61mKbcItVI8uPtWQm719uOGayXSF5dBsGqMvtisps+ikLCgHBualU36y8FdEVK5uv7hhiV4\ndRuXyWDxRqJeTaVWKGQ2WNTgpbU+qAi8p8RGQrJTDTfcKuLfXCaD3cs+IqsZj7wqFLVGyF3p4A5V\nV3WG9wtCklqucdY9rB/UPnePJINg8X7L66Cr6U+vmASUS0GLmMGDu6+6kEy9uq1/YrsB75hyKURU\nnZNFBsH2HsloIJVWq0likG42eKWdT5V8wghXPMF6JX75UWQw3XqXtRquZOrVjPsr6HL1/q7114EK\n+gzcRMSHZEW4TNDcGf1su/wEXaIMzuxu4uQ0q1ls2jrrIpD4ZlUXBA/hQBdiTX0RWisSvohMAqsP\nZ/2g+bMnjGPZ9ZxJbJnLbOdEY72fPdldXqEkk8B6lxXue2B+4qYGR+YVtggSG8M1X/goicaZ6eEP\nC1EmgYUo64MXLZWCnXS2Hdy6rraO8wjq9D3jriuJZjPMwom4KJPAQpSde43qUQMGtC34+0uwKSd6\nXW2iQfVXb0kmgYUL+XEpXpbUYFq70nre58jYq0RzLlL1MJkEtmLKH1P4x5Q/3xLkn0SdaJ4p4jKX\nYHfxduuZKeDq8VqmSTSPNAgyV2DNVWh9ZHCMubpJNI9EJZkLsONVynpk0O8DcNs20TyRFWXOwU4/\nwPoE1o95/gIC3CP+wsGPxAAAAABJRU5ErkJggg==\n",
       "prompt_number": 127,
       "text": "\n        \u239b1  1  \u2502  2\u239e\n\u256d\u2500\u256e1, 1 \u239c      \u2502 a \u239f\n\u2502\u2576\u2510     \u239c\u03bd  -\u03bd \u2502 \u2500\u2500\u239f\n\u2570\u2500\u256f2, 2 \u239c\u2500  \u2500\u2500 \u2502  2\u239f\n        \u239d2  2  \u2502 k \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2          "
      }
     ],
     "prompt_number": 127
    },
    {
     "cell_type": "code",
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKwAAAAuCAMAAACClwI5AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nwPoE1o95/gIC3CP+wsGPxAAAAABJRU5ErkJggg==\n",
       "prompt_number": 128,
       "text": "\n        \u239b1  1  \u2502  2\u239e\n\u256d\u2500\u256e1, 1 \u239c      \u2502 a \u239f\n\u2502\u2576\u2510     \u239c\u03bd  -\u03bd \u2502 \u2500\u2500\u239f\n\u2570\u2500\u256f2, 2 \u239c\u2500  \u2500\u2500 \u2502  2\u239f\n        \u239d2  2  \u2502 k \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         2          "
      }
     ],
     "prompt_number": 128
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAnBAMAAABZI83uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+3OGqwgAAAAASUVORK5CYII=\n",
       "prompt_number": 129,
       "text": "\nbesselj(\u03bd, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2      \n       r       "
      }
     ],
     "prompt_number": 129
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 130
    },
    {
     "cell_type": "code",
     "input": "besselj(nu+1, a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAnBAMAAACxo3jFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npORthDFrQGIAAAAASUVORK5CYII=\n",
       "prompt_number": 131,
       "text": "\nbesselj(\u03bd + 1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         r         "
      }
     ],
     "prompt_number": 131
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAABBCAMAAAD2WPjiAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n/+3zB2YkOkJkojheAAAAAElFTkSuQmCC\n",
       "prompt_number": 132,
       "text": "\n        \u239b 1/2     1/2   \u2502  2\u239e\n\u256d\u2500\u256e1, 1 \u239c               \u2502 a \u239f\n\u2502\u2576\u2510     \u239c\u03bd   1    \u03bd   1 \u2502 \u2500\u2500\u239f\n\u2570\u2500\u256f2, 2 \u239c\u2500 + \u2500  - \u2500 - \u2500 \u2502  2\u239f\n        \u239d2   2    2   2 \u2502 k \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n              k              "
      }
     ],
     "prompt_number": 132
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAnBAMAAACxo3jFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npORthDFrQGIAAAAASUVORK5CYII=\n",
       "prompt_number": 133,
       "text": "\nbesselj(\u03bd + 1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         r         "
      }
     ],
     "prompt_number": 133
    },
    {
     "cell_type": "markdown",
     "source": "Trigonometric functions"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 134
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "sin(a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAnBAMAAACs3UZeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABWklEQVQ4Ed2SP0vDQBiHf2lrDkJKooKCKIguIgoKfoCAFB07+AG6CkLiB5BGR10c\nRRwyuRp0cbDksndwc9VFHEQUBcEK9b38OU/knDp5kOS59zneuws/4Odw5dSQJMGMJbLbAqeCsjZS\nAn0XCp6VNlRsVVaLYoUr1naViUDTUwrWEmDMjFpH0Wpv+JGEQ4+dzqHSveoCz0ADZtCO2CcS2qZN\n9gQP2E74JnADrHs1OBFexAsd6hXiCWuLoLUHtFX/WIgPOB5wCdSb2APuqQn2geDcj6XdARKPvQO7\nwlLnCQxxaelUZ4ERTrOesHSqMdge2besczVCPb5o8co1SesQmExTy185fZ33l+k2HLW78a3YbpI1\nXNFAGSxUJr/+JDYU21A4RzOQJdaSWALjJcH+XihrA4T+X2OA+/y3VnnIdbfKQ66zech1Ng+51mYh\n19k85DqbhVwnkYVctV+QsmsDq7FoPAAAAABJRU5ErkJggg==\n",
       "prompt_number": 135,
       "text": "\nsin(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   r    "
      }
     ],
     "prompt_number": 135
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAb8AAADFCAMAAADzAwxTAAAASFBMVEX///8AAAAAAAAAAAAAAAAA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\nqqweoklx0t90Od1JX/Arm/0fVJ0R/58cNoUAAAAASUVORK5CYII=\n",
       "prompt_number": 136,
       "text": "\n\u23a7                                                      2 \u25021 \u2502    \n\u23aa                       0                         for a \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa                                                        \u2502 2\u2502    \n\u23aa                                                        \u2502k \u2502    \n\u23aa                                                                \n\u23aa                                                      \u2502 2\u2502      \n\u23aa                      1                               \u2502k \u2502      \n\u23aa                 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                     for \u2500\u2500\u2500\u2500 < 1  \n\u23aa                          1/2                           2       \n\u23aa                 \u239b 2    2\u239e                             a        \n\u23aa                 \u239da  - k \u23a0                                      \n\u23aa                                                                \n\u23aa  ___ \u23a7                              2 \u25021 \u2502                     \n\u23aa\u2572\u2571 \u03c0 \u22c5\u23aa           0             for a \u22c5\u2502\u2500\u2500\u2502 < 1                 \n\u23aa      \u23aa                                \u2502 2\u2502                     \n\u23aa      \u23aa                                \u2502k \u2502                     \n\u23aa      \u23aa                                                         \n\u23a8      \u23aa                              \u2502 2\u2502                       \n\u23aa      \u23aa          k                   \u2502k \u2502                       \n\u23aa      \u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500    for \u2500\u2500\u2500\u2500 < 1                   \n\u23aa      \u23aa              ________          2                        \n\u23aa      \u23aa             \u2571      2          a                         \n\u23aa      \u23a8  ___       \u2571      k                                     \n\u23aa      \u23aa\u2572\u2571 \u03c0 \u22c5a\u22c5   \u2571   1 - \u2500\u2500                                    \n\u23aa      \u23aa          \u2571         2                                    \n\u23aa      \u23aa        \u2572\u2571         a                                     \n\u23aa      \u23aa                                                         \n\u23aa      \u23aa        \u239b         \u2502  2\u239e                                  \n\u23aa      \u23aa\u256d\u2500\u256e1, 1 \u239c1/2  1/2 \u2502 a \u239f                                  \n\u23aa      \u23aa\u2502\u2576\u2510     \u239c         \u2502 \u2500\u2500\u239f     otherwise                    \n\u23aa      \u23aa\u2570\u2500\u256f2, 2 \u239c1/2   0  \u2502  2\u239f                                  \n\u23aa      \u23aa        \u239d         \u2502 k \u23a0                                  \n\u23aa      \u23a9                                                         \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n\u23a9                       k                                        "
      }
     ],
     "prompt_number": 136
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "sin(a*r)/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAnBAMAAACs3UZeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nd86Lr4B9AAAAAElFTkSuQmCC\n",
       "prompt_number": 137,
       "text": "\nsin(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    2   \n   r    "
      }
     ],
     "prompt_number": 137
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATUAAABuCAMAAABx/3S2AAAARVBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 138,
       "text": "\n\u23a7              \u239ba\u239e                 2 \u25021 \u2502    \n\u23aa          asin\u239c\u2500\u239f            for a \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa              \u239dk\u23a0                   \u2502 2\u2502    \n\u23aa                                    \u2502k \u2502    \n\u23aa                                            \n\u23aa                                  \u2502 2\u2502      \n\u23aa             \u03c0                    \u2502k \u2502      \n\u23aa             \u2500                for \u2500\u2500\u2500\u2500 < 1  \n\u23aa             2                      2       \n\u23a8                                   a        \n\u23aa                                            \n\u23aa              \u239b       \u2502  2\u239e                 \n\u23aa  ___ \u256d\u2500\u256e1, 1 \u239c 1   1 \u2502 a \u239f                 \n\u23aa\u2572\u2571 \u03c0 \u22c5\u2502\u2576\u2510     \u239c       \u2502 \u2500\u2500\u239f                 \n\u23aa      \u2570\u2500\u256f2, 2 \u239c1/2  0 \u2502  2\u239f                 \n\u23aa              \u239d       \u2502 k \u23a0                 \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n\u23aa             2                              \n\u23a9                                            "
      }
     ],
     "prompt_number": 138
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "b = Symbol(\"b\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 139
    },
    {
     "cell_type": "code",
     "input": "sin(a*r)/(b**2+r**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAoBAMAAABdi/SLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAACBElEQVQ4Ea2SsWsUQRTGv72sO2HYy10iKESNhzYiCSTgH7AgQQshV1haXCUogduk\nVbnVRjRNIBASsVgQbIxm1cbCsHNFTJErgo3tWSgKIZwkICSB9c3u7DgbwUYf7Jvvfb/ZNzM7CxSj\nqktLKy2cSEvWVfKMn3tDuaBxVOnzmgYG7dOuMkvCoG7VKKR0PMPg44B17jhfCi8fDG4TqNDjti+g\n1HnfAXrAJBy/FbJDxLRMi+hTfMNMLG4Bn4Crno1KiB8yYY16BdjBlTHQ3DlaKnkiwT4qHvAOKNfx\nAPhKTfAY8N80I03vAbHHfgL3JaXOwzgmNKVdrfpWUGMHktKuTsD1iO6lnftClKO3DVHaIsjngdPt\nNm9eer57sTlBpxGwP5+cjtw6UasqGxjBAqP440viukEnDZ1Jx9cWa2iZCyZyBff3RO39R5H8Lf59\nnZUP4dEmvHNbWUxwuq1i3MVLeeBn++A9uqMjsZpemkQIsws1J3C0IqqtcUoD9ZTwWjpk6YscnC6l\nEamAUjUbZba6MpdrgC2YlAV6Tf6WiAdvYOXOsElHNzuCb7wOyTvrlV8kSc+g7NShMx8nibRuol+k\niFK2rj27lS1EzkdU1F6+Ly88Wl4ka6BBKQ066VSUF2rP/Wo6wHrUOw9F0++QeYH9MIf5ica0gVfr\nvi7Uu+vaKAi7VijxC6dLlntvxZ4aAAAAAElFTkSuQmCC\n",
       "prompt_number": 140,
       "text": "\nsin(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2 \nb  + r  "
      }
     ],
     "prompt_number": 140
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 141
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "input": "cos(a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAAnBAMAAAChwzYZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 142,
       "text": "\ncos(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   r    "
      }
     ],
     "prompt_number": 142
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAb8AAADQCAMAAACgGJ97AAAAQlBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 143,
       "text": "\n\u23a7                      1                               2 \u25021 \u2502    \n\u23aa              \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                 for a \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa                      __________                        \u2502 2\u2502    \n\u23aa                     \u2571    2                             \u2502k \u2502    \n\u23aa                    \u2571    a                                      \n\u23aa              k\u22c5   \u2571   - \u2500\u2500 + 1                                 \n\u23aa                  \u2571       2                                     \n\u23aa                \u2572\u2571       k                                      \n\u23aa                                                                \n\u23aa                                                      \u2502 2\u2502      \n\u23aa                                                      \u2502k \u2502      \n\u23aa                       0                          for \u2500\u2500\u2500\u2500 < 1  \n\u23aa                                                        2       \n\u23aa                                                       a        \n\u23aa                                                                \n\u23aa  ___ \u23a7          1                   2 \u25021 \u2502                     \n\u23aa\u2572\u2571 \u03c0 \u22c5\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   for a \u22c5\u2502\u2500\u2500\u2502 < 1                 \n\u23aa      \u23aa            __________          \u2502 2\u2502                     \n\u23aa      \u23aa           \u2571    2               \u2502k \u2502                     \n\u23a8      \u23aa  ___     \u2571    a                                         \n\u23aa      \u23aa\u2572\u2571 \u03c0 \u22c5   \u2571   - \u2500\u2500 + 1                                    \n\u23aa      \u23aa        \u2571       2                                        \n\u23aa      \u23aa      \u2572\u2571       k                                         \n\u23aa      \u23aa                                                         \n\u23aa      \u23aa                              \u2502 2\u2502                       \n\u23aa      \u23a8                              \u2502k \u2502                       \n\u23aa      \u23aa           0              for \u2500\u2500\u2500\u2500 < 1                   \n\u23aa      \u23aa                                2                        \n\u23aa      \u23aa                               a                         \n\u23aa      \u23aa                                                         \n\u23aa      \u23aa        \u239b         \u2502  2\u239e                                  \n\u23aa      \u23aa\u256d\u2500\u256e1, 1 \u239c1/2  1/2 \u2502 a \u239f                                  \n\u23aa      \u23aa\u2502\u2576\u2510     \u239c         \u2502 \u2500\u2500\u239f     otherwise                    \n\u23aa      \u23aa\u2570\u2500\u256f2, 2 \u239c 0   1/2 \u2502  2\u239f                                  \n\u23aa      \u23aa        \u239d         \u2502 k \u23a0                                  \n\u23aa      \u23a9                                                         \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n\u23aa                       k                                        \n\u23a9                                                                "
      }
     ],
     "prompt_number": 143
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "input": "(1-cos(a*r))/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG0AAAAnBAMAAAD3FdtvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny7f8XNQdl4r833IJlm9D0A+eeJ5mKzqtUQAAAABJRU5ErkJggg==\n",
       "prompt_number": 144,
       "text": "\n-cos(a\u22c5r) + 1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       2     \n      r      "
      }
     ],
     "prompt_number": 144
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 145
    },
    {
     "cell_type": "markdown",
     "source": "6"
    },
    {
     "cell_type": "code",
     "input": "cos(a*r)/(b**2+r**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAAoBAMAAABQlYTMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n02fTnYsmyMWcdnMU9A8cYovJEOunyAAAAABJRU5ErkJggg==\n",
       "prompt_number": 146,
       "text": "\ncos(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2 \nb  + r  "
      }
     ],
     "prompt_number": 146
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 147
    },
    {
     "cell_type": "markdown",
     "source": "7"
    },
    {
     "cell_type": "code",
     "input": "cos(a**2*r**2/2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGIAAAAsBAMAAABs2XAhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nl7jwH8jSl+4MphqFAAAAAElFTkSuQmCC\n",
       "prompt_number": 148,
       "text": "\n   \u239b 2  2\u239e\n   \u239ca \u22c5r \u239f\ncos\u239c\u2500\u2500\u2500\u2500\u2500\u239f\n   \u239d  2  \u23a0"
      }
     ],
     "prompt_number": 148
    },
    {
     "cell_type": "code",
     "input": "#hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 149
    },
    {
     "cell_type": "markdown",
     "source": "Exponential functions"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")\nn = Symbol(\"n\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 150
    },
    {
     "cell_type": "code",
     "input": "r**(n-Rational(1,2))*exp(-a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAVBAMAAAAX07VdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA/ElEQVQoFWNgQAN30PjYuKxZ2ETRxbrQBbDxB6EijuwAbC4FijklBx3ALqWkbqIu\nmACWE9jE8AGrIi4Ff0ZuAYgilg+8GxiiVq1aFcDw//9/iHow197hHoMnVDtjAc8BGFML2U3SDDoM\n4lAZLgW+S1CmE8NkKAtEtTHsCFgEpBmVjJTYDHg0oFLXGCqArKAoAzCfK1RLoBjIcrVXSIcqAFEs\nDFMYGNgPMCsgiTEwtEgyOCAL8ADddJKBB2ISXGIZnAVmCALJ70HBqIIMW1H4HAI3GJh/oAgBOSyf\nUUTs/38BCzmgiDJOQOGCOYUMrA0ookwHULhgjleoMoogANpCMi2KoUbQAAAAAElFTkSuQmCC\n",
       "prompt_number": 151,
       "text": "\n n - 1/2  -a\u22c5r\nr       \u22c5\u212f    "
      }
     ],
     "prompt_number": 151
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAAuCAMAAAAGNLI+AAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nbwB/b+jP6f0/5fQvncE+WzxSwDcAAAAASUVORK5CYII=\n",
       "prompt_number": 152,
       "text": "\n                          \u239bn   3  n   5 \u2502  2  \u2148\u22c5\u03c0\u239e\n -n - 3/2             \u250c\u2500  \u239c\u2500 + \u2500, \u2500 + \u2500 \u2502 k \u22c5\u212f   \u239f\na        \u22c5\u0393(n + 3/2)\u22c5 \u251c\u2500  \u239c2   4  2   4 \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n                     2\u2575 1 \u239c             \u2502     2  \u239f\n                          \u239d     1       \u2502    a   \u23a0"
      }
     ],
     "prompt_number": 152
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, 0)",
     "language": "python",
     "outputs": [],
     "prompt_number": 153
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 154
    },
    {
     "cell_type": "code",
     "input": "(1-exp(-a*r))*r**(-2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAmBAMAAACCH1CXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAA+UlEQVQ4EWNgwAnSMhszyxWA0ozK2NVwJ8izcxUAVbCEfcdQcXr37t0H+gX8GGQh\nUpgqwOJVDKkMlXhVGDEsPrANq4pjpxvA4txnsgtmYFPBGcCRABFHkKjuiGVgbUDIQVhgFZ0zgWAa\nUODnsRPoChhQzOD4jCHPgKqC8QcDgwC6IhQzGKYz8BigquCI/LUZWUT6TAYyd8Sx/xMCtAkRtyQH\n/AYzJrBcxK+C5QMDtiSCrMkBlETwA+4A/PIMDKcJKWBOYMSthD2tLY3BTeIIbhUy/QmaHP//f8Ct\nwrAWM52jqd6ExsfkrsEUQhUhHFbsF1B1YPLYsIUmAGwCWtPpFhg7AAAAAElFTkSuQmCC\n",
       "prompt_number": 155,
       "text": "\n     -a\u22c5r\n1 - \u212f    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     2   \n    r    "
      }
     ],
     "prompt_number": 155
    },
    {
     "cell_type": "code",
     "input": "h = hankel_transform(_, r, k, nu)\nh",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAg8AAABDBAMAAADpDvuvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvr6g6fC5NJYAAAAASUVORK5CYII=\n",
       "prompt_number": 156,
       "text": "\n    \u239b                       \u03bd                     \u239e                     -\u03bd\n    \u239c   \u239b      ________    \u239e                      \u239f \u239b      ________    \u239e  \n    \u239c   \u239c     \u2571      2     \u239f                      \u239f \u239c     \u2571      2     \u239f  \n -\u03bd \u239c \u03bd \u239c    \u2571      k      \u239f              \u03bd       \u239f \u239c    \u2571      k      \u239f  \na  \u22c5\u239ca \u22c5\u239c   \u2571   1 + \u2500\u2500  + 1\u239f \u22c5\u0393(\u03bd + 1) - k \u22c5\u03bd\u22c5\u0393(\u03bd)\u239f\u22c5\u239c   \u2571   1 + \u2500\u2500  + 1\u239f  \n    \u239c   \u239c  \u2571         2     \u239f                      \u239f \u239c  \u2571         2     \u239f  \n    \u239d   \u239d\u2572\u2571         a      \u23a0                      \u23a0 \u239d\u2572\u2571         a      \u23a0  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                \u03bd\u22c5\u0393(\u03bd + 1)                                "
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "input": "h.subs(nu, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAANBAMAAACAxflPAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMARGZUmbvN3atbSixTAAAAGUlEQVQIHWNggANGBQiT\nmaq0a2hYaWgEAwBXFQOSs8FFyAAAAABJRU5ErkJggg==\n",
       "prompt_number": 157,
       "text": "nan"
      }
     ],
     "prompt_number": 157
    },
    {
     "cell_type": "code",
     "input": "# Too bad, the solution for nu = 0 is very simple: asinh( a / k )",
     "language": "python",
     "outputs": [],
     "prompt_number": 158
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(h, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 159
    },
    {
     "cell_type": "markdown",
     "source": "Bessel functions"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 160
    },
    {
     "cell_type": "code",
     "input": "(1-besselj(0, a*r))/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAnBAMAAADp3OvcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\na7DgNgEp6Tryi5f46PoQdDeKfAG5CJG8jzxhTAAAAABJRU5ErkJggg==\n",
       "prompt_number": 161,
       "text": "\n-besselj(0, a\u22c5r) + 1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n          2         \n         r          "
      }
     ],
     "prompt_number": 161
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANsAAABBCAMAAABvoCFXAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nHzIIGYFxJQAAAABJRU5ErkJggg==\n",
       "prompt_number": 162,
       "text": "\n \u239b          \u239b1  1  \u2502  2\u239e    \u239e\n \u239c  \u256d\u2500\u256e1, 1 \u239c      \u2502 k \u239f    \u239f\n-\u239c\u03bd\u22c5\u2502\u2576\u2510     \u239c\u03bd  -\u03bd \u2502 \u2500\u2500\u239f - 2\u239f\n \u239c  \u2570\u2500\u256f2, 2 \u239c\u2500  \u2500\u2500 \u2502  2\u239f    \u239f\n \u239d          \u239d2  2  \u2502 a \u23a0    \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n             2\u22c5\u03bd             "
      }
     ],
     "prompt_number": 162
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 163
    },
    {
     "cell_type": "code",
     "input": "f = besselj(1, a*r)/r\nf",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAnBAMAAAC24abQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZonvmSLNu3ZEMt2r\nEFSC55RrAAABRklEQVQ4Ed2SP0vDUBTFTyrWpCYxuCtFJ7ciOIoKOrVDQBDBod1cHR3TrZvFXfAb\nGAcXQYmTjil+gAouBQe7CCKI3vfue883vG6dvEPuueeXd/OHA9iVmMEzSokwNU60zXJxa4/FskHA\nSOlj1XOLVUsePrnFmcV8fnbwxV7Yt1itI4dKg70Fan73DXHvsAeMpTlfMBtSO0UTg3b2CGxKc5hg\nLiW1D3g5dnHxihK4kew9jVcFOwJm6rgHWsK+laxJnmDPQLsffQPXwqad3gbWFaN3uSq9/CX6EGwM\nhJ3KnWLVSzp/UGRxg1DtBJjtrpSK+RmCnaWn1K8T8xK6yJLPi3I9Utf/TJ3DmsXOtQ7OHoQMabuq\nqNBKG5kx/L/bjDcV8TO5prL/HyzhzLo/hDPrZpxZN+PMTmAys27GmXUzmVk3gsysxX4BFB1YkoPT\nwwIAAAAASUVORK5CYII=\n",
       "prompt_number": 164,
       "text": "\nbesselj(1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       r       "
      }
     ],
     "prompt_number": 164
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbkAAACwCAMAAACl8uBmAAAATlBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
       "prompt_number": 165,
       "text": "\n\u23a7                                                 2 \u25021 \u2502    \n\u23aa                    0                       for a \u22c5\u2502\u2500\u2500\u2502 < 1\n\u23aa                                                   \u2502 2\u2502    \n\u23aa                                                   \u2502k \u2502    \n\u23aa                                                           \n\u23aa                                                 \u2502 2\u2502      \n\u23aa                    1                            \u2502k \u2502      \n\u23aa                    \u2500                        for \u2500\u2500\u2500\u2500 < 1  \n\u23aa                    a                              2       \n\u23aa                                                  a        \n\u23aa                                                           \n\u23aa\u23a7                               2 \u25021 \u2502                     \n\u23aa\u23aa           0              for a \u22c5\u2502\u2500\u2500\u2502 < 1                 \n\u23aa\u23aa                                 \u2502 2\u2502                     \n\u23aa\u23aa                                 \u2502k \u2502                     \n\u23a8\u23aa                                                          \n\u23aa\u23aa                               \u2502 2\u2502                       \n\u23aa\u23aa           k                   \u2502k \u2502                       \n\u23aa\u23aa           \u2500               for \u2500\u2500\u2500\u2500 < 1                   \n\u23aa\u23a8           a                     2                        \n\u23aa\u23aa                                a                         \n\u23aa\u23aa                                                          \n\u23aa\u23aa        \u239b          \u2502  2\u239e                                  \n\u23aa\u23aa\u256d\u2500\u256e1, 1 \u239c1/2  1/2  \u2502 a \u239f                                  \n\u23aa\u23aa\u2502\u2576\u2510     \u239c          \u2502 \u2500\u2500\u239f     otherwise                    \n\u23aa\u23aa\u2570\u2500\u256f2, 2 \u239c1/2  -1/2 \u2502  2\u239f                                  \n\u23aa\u23aa        \u239d          \u2502 k \u23a0                                  \n\u23aa\u23a9                                                          \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n\u23aa                    k                                      \n\u23a9                                                           "
      }
     ],
     "prompt_number": 165
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(f, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKEAAABBCAMAAACOyzQGAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAEImrmSJ2\n3UTvVLvNMmZAoe8m1AUAAATTSURBVGgF7VrpktsgDAYM+MSt3/9lK05LRhAn2U4z0/jHWgYkfQh0\nQFaIf/lI9/PazfSTMp1ipck3lJhNsjJfbGwgFHp/UaCQPwtQtBCKdXkR4oYZF70r/N2XKQdm6ZoI\n5Wj64hq9akAd0wYfK6MWjSmkU/ZgZtNEKKax8D5BuBlvws3vld3DvPc8h1Cs9p5YMmrFnieDxunA\noMno68eTCN19yUUT5XGHX2B33I5oNxAuFjkxsUcB0SVWvAvFFLBFnF223PkYoYSFnYuHTEfmvL7X\nMkYgUghJzfUXEDowAXK9mQ+KiyqzQKSfhZ3JXKL1Ik7S0frgbPjrdxyto8PBnkbOmBpreQUhNjkM\n28gig0l99Fju72cOYY42u044FLJb0wsbCCMiNKEQDn4o2uQUspCYyc3JA2ggrKwVIvZ2M2KDXE5f\ntuEs3Q5LNC3CIYEjim5SDVopY3xTA6Gi2xAGLmq3ZM7Q1nqMWo9RI+1xYELotmWSgzAHPCi+DmsR\nZ8fAq2avsIFwvZ89ityHREJoZ4WAFS5b4s2aKhYZJhAQTiEQI7BipI5SxLxFJITr5Lg0vBwp8OlM\nCL8V7DGAP2kwcCIThBltirdQYeaEEGx1COTFaUhOB9PB5ehq/FG1YFUv0hGhAXOovU6fJjnXijdn\n0VSh5lyxjH6VSDZssMtklSO6gNUbytXLdefm+TSEvdjcRyjS6h7ZBY4c2Dl1eU/kPpKyc+ODd80T\ns54PMug5pRxx788JoeuuI0V4SdmnzA7F8RAb0tLGS0oIdQrF0aWdtUPas4gUEE1pcMaBqAOLdNU8\nBOGltPEIkzdsYXWnwSP180jFPiJ9x8WXa20EDPtR8xCEgpY2IKLotIO2kL98OHE+jIfqGZNeXxnt\nP0jmiQ03/j5C6GMOFlMVK6ETVlimCI5I6MoWTyJqbVg2T9c81IaCljaws6pEHgXrM3sgcqaeXmvj\nUeHWmicjtNaCo19KGzhonOU+llOqSThrIlTrWWj40bU2LIOna56EUFtYuKq0EXupHIi8CXwlhWtE\nXg4BNGUT/uYHxxMRGshqjimdNFdOCGelgRTpa5tEJpV8Tb5vIVb5YuP2g3giwgWQ2HNrFUH0cJma\nQxEJxoXaJpOpR/LbVvnFN40NU3RR4uSJCCfIGeNUT/ISPqgQphZCNXlOUcATTitB5UXA9ZPlSWc9\nve9W04wA7C1HSZKr2kYodlf4Q7SsXQCkmLU8F2EnT/KU62ziN68wj61qGx/MGd8PEZ7c2WUJ7Tfi\n6SLcmK3Zlup7xtMUJWVbcMGbbsLx9BDSa6I+tNTrXS4+Z8pelcmFRu5svFmeHkJ00muIrJvH7G5n\n9p5v38KyPB2EOffWMDotSzky05TdYUFdHE8HIbnvRWL65JqNCMNQyu4zod6Kp43QoCsmJOERie6/\nccp+xJb7a542wieuW7L48A53NZ4iKZsMaX8wPE2E6ulIk/Uu8VBDU3bu7L85nhbCBRVVfal17+K3\n4iVl16OYFpangdC8AZBR/W0Sueb42Pd3jb4W+Frga4GvBf5XC1i7n+eZTzSCP+Dz1y0fgnb3x0H9\nybVT+OFqrO43PsR+ACNcmsZfBz8HFEESrglgpT/YiHDFZOC/Dc7rEzKBT/iY4J5NDpa5gnod3R9K\nEi1RTTxSWQAAAABJRU5ErkJggg==\n",
       "prompt_number": 166,
       "text": "\n        \u239b0  1  \u2502  2\u239e\n\u256d\u2500\u256e1, 1 \u239c      \u2502 k \u239f\n\u2502\u2576\u2510     \u239c\u03bd  -\u03bd \u2502 \u2500\u2500\u239f\n\u2570\u2500\u256f2, 2 \u239c\u2500  \u2500\u2500 \u2502  2\u239f\n        \u239d2  2  \u2502 a \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n         a          "
      }
     ],
     "prompt_number": 166
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAAnBAMAAAC24abQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZonvmSLNu3ZEMt2r\nEFSC55RrAAABRklEQVQ4Ed2SP0vDUBTFTyrWpCYxuCtFJ7ciOIoKOrVDQBDBod1cHR3TrZvFXfAb\nGAcXQYmTjil+gAouBQe7CCKI3vfue883vG6dvEPuueeXd/OHA9iVmMEzSokwNU60zXJxa4/FskHA\nSOlj1XOLVUsePrnFmcV8fnbwxV7Yt1itI4dKg70Fan73DXHvsAeMpTlfMBtSO0UTg3b2CGxKc5hg\nLiW1D3g5dnHxihK4kew9jVcFOwJm6rgHWsK+laxJnmDPQLsffQPXwqad3gbWFaN3uSq9/CX6EGwM\nhJ3KnWLVSzp/UGRxg1DtBJjtrpSK+RmCnaWn1K8T8xK6yJLPi3I9Utf/TJ3DmsXOtQ7OHoQMabuq\nqNBKG5kx/L/bjDcV8TO5prL/HyzhzLo/hDPrZpxZN+PMTmAys27GmXUzmVk3gsysxX4BFB1YkoPT\nwwIAAAAASUVORK5CYII=\n",
       "prompt_number": 167,
       "text": "\nbesselj(1, a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n       r       "
      }
     ],
     "prompt_number": 167
    },
    {
     "cell_type": "markdown",
     "source": "Algebraic functions"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 168
    },
    {
     "cell_type": "code",
     "input": "r**nu/(r**4+4*a**4)**(nu+Rational(1,2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI0AAAAeBAMAAADjgoB7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 169,
       "text": "\n              -\u03bd - 1/2\n \u03bd \u239b   4    4\u239e        \nr \u22c5\u239d4\u22c5a  + r \u23a0        "
      }
     ],
     "prompt_number": 169
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbUAAABICAMAAABGD5ucAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n2gL8yuN/JFZoQBRVpOYAAAAASUVORK5CYII=\n",
       "prompt_number": 170,
       "text": "\n -3\u22c5\u03bd               \u239b     \u03bd   1                 \u2502      \u239e\n \u2500\u2500\u2500\u2500               \u239c   - \u2500 + \u2500                 \u2502  4  4\u239f\n  2    -3\u22c5\u03bd \u256d\u2500\u256e3, 1 \u239c     4   2                 \u2502 a \u22c5k \u239f\n2    \u22c5a    \u22c5\u2502\u2576\u2510     \u239c                           \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n            \u2570\u2500\u256f1, 5 \u239c3\u22c5\u03bd  \u03bd  \u03bd   1  -\u03bd    \u03bd   1 \u2502   64 \u239f\n                    \u239c\u2500\u2500\u2500, \u2500, \u2500 + \u2500  \u2500\u2500, - \u2500 + \u2500 \u2502      \u239f\n                    \u239d 4   4  4   2  4     4   2 \u2502      \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      4\u22c5\u0393(\u03bd + 1/2)                      "
      }
     ],
     "prompt_number": 170
    },
    {
     "cell_type": "code",
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAbUAAABICAMAAABGD5ucAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAZomZu1R2\nIjLvq90QRM1AofVFaV4AAAqkSURBVHgB7VzpoqMqDMYNbV3m+v4vewOYQNjEattzZvRHqyQkIZ9s\nMSrEbzjavtzKI7zlUi/inIaLBH1PTD+X6e7qRzFqR3jLtF/KNdQvipsKffWi+PJq/TKVMpejJsQR\n3lL9l/G9jJpoqsuMOCNoKgftEBJ/KWqi7c54+6q6ywErjiBxhPeqthTLeb2viWksniaKzTnMWD8P\nVMkgMfjgZ3gPaHwT6wnUxDy+yahyscOjeFIDoRkkOn+8z/CWm/cuzjOoiVbumtV1cp9pV0qSoT20\nmMog8Q+hNqy7dzp0x0fS56cJBQZYHbNcn7pHVYs2yVxsdA814rXVf9LZqb4mSm71fnlfe9sjs5o1\no27hvF/dadlDzfL+yLNzqM3rbqOqxXXOLnuEYe6qJ8UCWMhiWqk8Ui9dpEd2DR3x/EuoiYc/iZMb\n7Mm4O4xa3tjZYxDzNsp6IQuZHXz7lg41ta7boU4B7IkmuUaxLaP6xa7r8MYs+nrZn/+sCew+tsW5\ns2Zn+JtgQb2/r2O9KVAHfdX2afK1YlvQy0GdbMGgRghvn/db+5p3Hzvtrv1xyBbMO+uRuRFi9Ks7\nos2p05sCmi6wcRgXtWn1N1nx2rbUGC7hVmNLEWDwUBuktKOyrf9Tztx5zfWIY98MTmWHW7DnuKqq\n9cq/Vvd3yhGsNzFV+mKonzTIujZ2O7dMIGkzvK37wBKO2gQ2H9sKBrreWrCP2lQtCrVuQR9tBcas\nkTZMVd1kxsIB7u8+Mw2p3sSQnepnU9d9r+XPtJl2Uasz8mJOQ8MfddhHOWp6DF1/Snw8bMs+alJo\n1ESHPsICLeypFtHqkNBhhhV8XzXm0OsUWgFI8L4d50wV+2t6k4usHLXP6ken9SrB+nBRa3cmVSvf\nnDHDPeLMgYT7dGI7A4/9y5e7qHX9hhpG/KhAWy5x7f9QA+ETkaVWddJg1c6igznOORi6sEqchINs\nu0Xyp3VSFqol/qzHaRe18dhihBvuWBI/bWgUidO/WbqHmloEmr62mNvdFmizO7wj9VBZ4yiKTVIT\nhHb0OlXYXZBG/9ibLLINShUATCerJ1jRQK/mIYvHIb96hpP2xEnF77EE15eK91CrpJSPBkYrHDCo\nwBg8sNG/9fvaAF4HMEQ/VrjZlrLqeNgZexMhO696BeO5JECdhk2PMX7pGR5notIZLKAlEJX+lBOL\nGr+PheNcvXRXG92uNjOJs5bv3UXkFHobWk4rCd1mFZdoOGpbb7LItn6X1TUDJF3VZf50DM9XGOTU\nV97SOV/jo1SLGldrnTtB0BV6C0xrctKdZCsw/JN7w7exUaVmfaRSgDUxPkf/am4O2SxOoLPzb312\nwzi1k6fM8CSXIvR6FZVl+SoxgVro3IV7n4x2updaTQRHxxdnei4aeVlQZ8VlxpqDlw3O+fBKNuqT\nJXLbylnfwQmTzTbNmIgWxt1ooR44d1pbGR0xVoKqi00FsKIe9BreNL9X0yAsDLkz/KvHhtqQHQMZ\narnwSjrqA4qzRG5YOes7OF1T430tdC5Ma/EIO6Gm5m/hzz16qHEw0g/EoCPnO1uzLWpwKYm3GOzD\n1wdVZSNkPrzi7hc4EnCVJXLuctZ3cFpT46iFzlW7NVj7k8uoNThCDm1VVRL33ET2T6A1PQR9fXQ9\ntkWPjLPZ/jk3rpycO4BNqSAgvYnPA1Pu4Lwc1oZyoeWcIWqwKTKeNAHWwLlqtzbOE1tZaDNxNfLQ\no+suanNTVdNTBVKyh3w2UnZiU0cN42Cjai3JDVYGoklAQIGCLJFXKGd9B6c1detrVQ0DoLJwC7AG\nzlX5WF3Dvab4j8fdVa2DB7lAVlVjZ1fs5ps0G6wMxJOAgAIFWSKvUM76Dk5r6oaa2lOp+x8DrNzY\nzJWeATP0S0jkAhigJ7uRf9gVJoZXEupIQIyeJp5It0sL9U1Icwbq6QYzqA0QpNVDYy7A6qvT1zPF\nnqLkawpZw+xzo9aOx6oZ8dWStoAJ8G1KE/mDAKiXZi0XWs4ZqCf9BrVK+R7628EAKxhQYfTYN+bK\na3TWoPqZRc1NQNiClVGtXtSH8+SInttyrOVCyzk99U4U1qCm8zLG7mCAVen3g1PcpouuNGoQ85+g\nRw02GhZ9kB5Lm0MzcjQRIfpuQznl/xGhicoRzrR6g5p+Xjl2xwKsWj0lyCSMuaB4u8VNzF82du0/\nscg1aoqkzSFJ5GgRYtptJHHvJKuRVQ450+oNakKlv5kob3GA1ahkq29mxeUX4a7DPkjH4A4oDdLm\ncjQ3cyuo6CWTOHJ22uZwhkJZ3Synh5rDu+VoTY3US+ryAKtR/pHFyNbOcNtR29HS+oKnzdlydZaj\ncWIk3Y6LYldeEp9D8zUWcu6o3/qao+fQadRvhyQUMwcxf5XxoHYr/IilzSFHjgbCgI3n23k3O4o5\n8B8Rmqgd4UyrP4naQrHjhDHvLR7d/pdOm4PnTHpvHkupAwOTRM9th9LtkkIDjyQ5PfXC6j+H2qG3\nIwJzzxe4D8X9tLnBmaF9GqxEbYglIJJd3G2H0u18oa5Ckq9PfE5L5eqFo/8capShZVV99mykJQpG\ndShtbqCnERTxIRoMhzRIBBVtC7jb9BgWXbbaGngWCHUUIo/5Dzgtmas3Q7jRfwo1zCWxij59Rul+\n8BTBJCU5FhCiERoGptXDpaAiyng93S4ilKxB6eY/wokMnnoBw8Pm8VOoYbIdqvnCf7s5IxbVQT/F\naIRalJhsSGG6XUwoWsNlxzg5B7va9J9Bred5PEz8py62l8NjUR1wiD5iNBW7SxMNKfJbmG4X04gK\nudQYJ+dgV6j/DGqLk1rAhH/yYtaZQbGoDj7BjdHo8W6UmLK/NN0uJhSt4bJjnJzDvSL9J1CraUJ3\nJX/8vMPUIGdZGBiRoxW89GPkHUq3y2pk9hVzWv2vo+YlgDNDPnphllq5qE6OBgtqnTm4b/KRdLti\noeXq3XS/l1Hr7SPJ/QbfHLcHbg9c5QEMSN//v8gDV4F/y7k9cHvgn/JAb2PL2XabpOgf86nMrK1/\nPbHHLWC+pZQUDfm29/F1D7Q2NyVvCyaPFVfIi7upJzxQ/pU9RE29Ensfn/KAGgqH5rE+6/o50vtZ\n5Z+QQtTET/jA6aec9nU9i85e3r4BgF+O6WxKs2dgkMJNqEVfuPRq35fXeECaZ/EbaojJMxkV9x4u\nOxnk+Hm2a+y6pWQ8MMwqM1Z9U8isPrbUIfN9mli9NGq7r7/GxN1lr3hA6m+YWNQ2GVu63n4Kt5v3\n//gJzxpfccJvqwNdi/c104AeVyVHUriFTTv6bW74XfaqLzzFUNOpWKopYbJ3MELaFi9uZqYtvs+u\n9YB+WTmLGkv2jqRwY3Bf23Wjdi08CWldDcc41jAf4WrEcOIIeSiFW+x9QiVhxV38ggfYyn+rj0me\nkfRyb4S0GdywB7hXIy/4/7Uq5h1h3tcQgMgXWzlqTgY3vK0Tvifymkl3rR0PDM26NoOKaLXmDQnD\nr1ch0HvCL7Zy1NwMcv3u8Y66m/xOD1TJULCXwm0zuEW6zjstvWVbDzjfxrCFiTPMIE++KpCodxdf\n7oGm+CEnZXDHXlq93K5bYNYDyaC/V4syuJ0PDnks9+XHPDAno/7MBMrgLuRnle+Lyz3QlWT7UAa5\n2gL8yuN/JFZoQBRVpOYAAAAASUVORK5CYII=\n",
       "prompt_number": 171,
       "text": "\n -3\u22c5\u03bd               \u239b     \u03bd   1                 \u2502      \u239e\n \u2500\u2500\u2500\u2500               \u239c   - \u2500 + \u2500                 \u2502  4  4\u239f\n  2    -3\u22c5\u03bd \u256d\u2500\u256e3, 1 \u239c     4   2                 \u2502 a \u22c5k \u239f\n2    \u22c5a    \u22c5\u2502\u2576\u2510     \u239c                           \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n            \u2570\u2500\u256f1, 5 \u239c3\u22c5\u03bd  \u03bd  \u03bd   1  -\u03bd    \u03bd   1 \u2502   64 \u239f\n                    \u239c\u2500\u2500\u2500, \u2500, \u2500 + \u2500  \u2500\u2500, - \u2500 + \u2500 \u2502      \u239f\n                    \u239d 4   4  4   2  4     4   2 \u2502      \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                      4\u22c5\u0393(\u03bd + 1/2)                      "
      }
     ],
     "prompt_number": 171
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALgAAAA1BAMAAAAAMzzyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 172,
       "text": "\n                    -\u03bd - 1/2\n          \u239b   4    \u239e        \n -3\u22c5\u03bd - 2 \u239c4\u22c5a     \u239f        \nr        \u22c5\u239c\u2500\u2500\u2500\u2500 + 1\u239f        \n          \u239c  4     \u239f        \n          \u239d r      \u23a0        "
      }
     ],
     "prompt_number": 172
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "r**(nu+2) / (r**4+4*a**4)**(nu+Rational(1,2))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAeBAMAAADA2dGuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWeAPQgGGqPhOt1MAAAAASUVORK5CYII=\n",
       "prompt_number": 173,
       "text": "\n                  -\u03bd - 1/2\n \u03bd + 2 \u239b   4    4\u239e        \nr     \u22c5\u239d4\u22c5a  + r \u23a0        "
      }
     ],
     "prompt_number": 173
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAABJCAMAAAANIH8qAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nmWwK3NYC9Y6zmubem1js/fTzf6Vubg4t+c5WAAAAAElFTkSuQmCC\n",
       "prompt_number": 174,
       "text": "\n   3\u22c5\u03bd                       \u239b       -\u03bd                      \u2502      \u239e\n - \u2500\u2500\u2500 - 1                   \u239c       \u2500\u2500                      \u2502  4  4\u239f\n    2       -3\u22c5\u03bd + 2 \u256d\u2500\u256e3, 1 \u239c       4                       \u2502 a \u22c5k \u239f\n2         \u22c5a        \u22c5\u2502\u2576\u2510     \u239c                               \u2502 \u2500\u2500\u2500\u2500\u2500\u239f\n                     \u2570\u2500\u256f1, 5 \u239c3\u22c5\u03bd   1  \u03bd  \u03bd   1  -\u03bd    \u03bd   1 \u2502   64 \u239f\n                             \u239c\u2500\u2500\u2500 - \u2500, \u2500, \u2500 + \u2500  \u2500\u2500, - \u2500 + \u2500 \u2502      \u239f\n                             \u239d 4    2  4  4   2  4     4   2 \u2502      \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                              \u0393(\u03bd + 1/2)                             "
      }
     ],
     "prompt_number": 174
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARoAAAA1BAMAAABsEDneAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAAE40lEQVRYCe1YTYgcRRR+ne7t7tmZ2RkQoujCDlGEKIFRJJ4kDUIuojtEV1CUjEIG\nBEOGHIIo4uDBgz9ZiRBZiGEQUZQ9LP4c1l2kFRUhQVbQUzysoIfcJhpNFGGselXVVdVdtT0705E9\npGFr3s/3Xn1b3V1fzQDsmMtr7RgqhMhXKzuJjXudjfV2XF8b69LA/742U1GazNQAbuOxu5fTyWvs\n3xelJ/A2w410rFA/bNraOd9HNHUDy59bW1uL/VY5rhBjHWrD4ZAlCh1P1m3t/JmIpsoxHdlVar4g\nzBPrxHJuFW4xn84v1j7PMTbwtkTUZmPuOO1qD7xHr8hcEdZM19bF6XI2N/cSyKvfCdNrlOgtLpjN\nIdE+81npHP0Eg24/kyOBepXe4mLZeD+YZuKx6YgZp42YJ2i0WDallnGmoPNKB5xj/Bmdp6uQvoIu\njRTLxjgRwOxie6+c3kj5Xq9ZOJtTck7VuutFUNbDHag5ZofDIUUUujbO5ew8GPlYi/+seaozMhtF\nZbyOUBfaySV/B5+OyAj+Bh0N17IW+0LzVGdkNorK7IYHlYUvkz20zV6mmYbaWtrOVWkT64BSrSVG\nvlNcZbB4HmpN2YWw8QbwBw3MdWVYtYIN1YOaBRY+9g/bkjQ0OodhV4QGG4jKJPrvwVwPo4ghbCBi\n//1LdQZOj35Li0zrrpZLOS9z/yF4Q80QlVH0/zOWQgxlA2Wc4EkM52rfrjdZ+QjjRWAafwSoqpIL\nXaoyVP/xIACVFZYimFv23LHnToBzGDhLx0T7vC5DZUZ/kAlZAr7Ypt4K+JQIpCoj9X8/3z4QQ9fG\nbTsU9i4dkl0jiNHLDoH+UGcAISkMIhq+f5WO5Lp0nn2KcTpK9N976vUIw4ihbA7e9DUN/IRR8WZY\n2Xj4yDOscSR35Tgm2pxNuNwjvv/s8xxOVCbR/0VyTqPSwzCEDdlJcfH/ZWC+a1jZhHnbSjkOGrSV\nD2JtsPFp+Bw/s4OUnimZ5LPksXE4a1mYttbZ0tyjsan0YS0N5L4uPSwo/uc8NvCXuSk5G/PTcWmT\nIpxYY1NtwQfmQgBdehAlngdks7p09sOlUyROD+DyQuCftqYifhzf52Bh4aOHRQhgselckp5u6dKD\nOY0NeS1ivUJ6eWyChjjLizec1s7XK33LppGSHpxp0juV0CVPDS4OwFISA6j2Tjbaiq+YKenBjHg6\nc5+bnKc4JDtduUlbPjD8VM7prn95e0+6qpWSHpbCWRLtM9ypRxDncLpqv2tg6++Km7nJ7o84aWjb\n/b5ZzZRMwPKCpVYo8eyvCHAtyuD3vJx7aOlvDqvPnIq4yJ02Y1MZqElpV9uwT3oTW0fNHYQSBz3G\nptw348i553DdnBonajliCiW+ERibUsPa/B1rZvuJuaaxhiox1d9vOxcQUIuNMBJ09PfABhstbj5i\nohIz/X0P+xzo2dqVVmyZMeJTfVMRVWKmv7t/x7V53ITC2BFrZoyEZ3p1UYk1/f3N1jpgR0hbervx\n9w0FqMSq/npWJd5PzlcFXuKkmGq5V9Nf61cG78z5Z1KVE7nTsbF8SdNf/mUsi6wOh39no+NHTMqO\nSqzq76Hx+2+z8kw+Ptzq96/88u0glJ8YbWUz5i3SBp8k7m7mVi/kIooDkEPk1pf2FXZr6ORZw4FP\nb/qa7iref5clKOm9mUTOAAAAAElFTkSuQmCC\n",
       "prompt_number": 175,
       "text": "\n                               -\u03bd - 1/2\n                     \u239b      4 \u239e        \n -\u03bd  -4\u22c5\u03bd - 2  \u03bd + 2 \u239c     r  \u239f        \n4  \u22c5a        \u22c5r     \u22c5\u239c1 + \u2500\u2500\u2500\u2500\u239f        \n                     \u239c       4\u239f        \n                     \u239d    4\u22c5a \u23a0        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   2                   "
      }
     ],
     "prompt_number": 175
    },
    {
     "cell_type": "code",
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARoAAAA1BAMAAABsEDneAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nb/qa7iref5clKOm9mUTOAAAAAElFTkSuQmCC\n",
       "prompt_number": 176,
       "text": "\n                               -\u03bd - 1/2\n                     \u239b      4 \u239e        \n -\u03bd  -4\u22c5\u03bd - 2  \u03bd + 2 \u239c     r  \u239f        \n4  \u22c5a        \u22c5r     \u22c5\u239c1 + \u2500\u2500\u2500\u2500\u239f        \n                     \u239c       4\u239f        \n                     \u239d    4\u22c5a \u23a0        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   2                   "
      }
     ],
     "prompt_number": 176
    },
    {
     "cell_type": "code",
     "input": "expand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMkAAABIBAMAAACuBMxLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 177,
       "text": "\n                          -\u03bd\n                \u239b      4 \u239e  \n -\u03bd  -4\u22c5\u03bd  2  \u03bd \u239c     r  \u239f  \n4  \u22c5a    \u22c5r \u22c5r \u22c5\u239c1 + \u2500\u2500\u2500\u2500\u239f  \n                \u239c       4\u239f  \n                \u239d    4\u22c5a \u23a0  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n              __________    \n             \u2571       4      \n      2     \u2571       r       \n   2\u22c5a \u22c5   \u2571   1 + \u2500\u2500\u2500\u2500     \n          \u2571           4     \n        \u2572\u2571         4\u22c5a      "
      }
     ],
     "prompt_number": 177
    },
    {
     "cell_type": "markdown",
     "source": "Exponential functions"
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 178
    },
    {
     "cell_type": "code",
     "input": "exp(-a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAmBAMAAAC8Oj/ZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAqUlEQVQoFWNggAMmpZBJUevgXBDDiPPDglt8UCFGY2NjQ5af3Ac4C1AUcV1gSuAK\nQBHieMDbyKwAFmJUNIFIGQpZsTqAmWEMThAhOMlWwJAC50AY/IGKDmhC/g1oAgwM8x0YONEEWRcw\nSKAJcRoKOaAJDQruf3TwYVA4izRHLFJRUkDVwSnyiQ81PTBwzCpAjycG5geo2oA8XgEMIWAMo4Pl\n6AIMDGoIIQBUNSlnoAeAjgAAAABJRU5ErkJggg==\n",
       "prompt_number": 179,
       "text": "\n -a\u22c5r\n\u212f    \n\u2500\u2500\u2500\u2500\u2500\n  r  "
      }
     ],
     "prompt_number": 179
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAa0AAABEBAMAAADARHf6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWhaErHB3bV2eHfevcHvc82v8A24a/OT2YTcSAAAAAElFTkSuQmCC\n",
       "prompt_number": 180,
       "text": "\n                               -\u03bd - 1                              \n           \u239b      ________    \u239e       \u239b         ________          \u239e\n           \u239c     \u2571      2     \u239f       \u239c        \u2571      2           \u239f\n -\u03bd - 1  \u03bd \u239c    \u2571      k      \u239f       \u239c 2     \u2571      k      2    2\u239f\na      \u22c5k \u22c5\u239c   \u2571   1 + \u2500\u2500  + 1\u239f      \u22c5\u239ca \u22c5   \u2571   1 + \u2500\u2500  + a  + k \u239f\n           \u239c  \u2571         2     \u239f       \u239c     \u2571         2           \u239f\n           \u239d\u2572\u2571         a      \u23a0       \u239d   \u2572\u2571         a            \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                               2    2                              \n                              a  + k                               "
      }
     ],
     "prompt_number": 180
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 181
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "exp(-a*r)/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAmBAMAAAC8Oj/ZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAzklEQVQoFWNggAMmpZBJUevgXBDDiPPDglt8UCFGY2NjQ5af3Ac4C1AUcV1gSuAK\nQBHieMDbyKwAFmJUNIFIGQpZsTqAmWEMThAhOMlWwJAC50AY/IGKDmhC/g1oAgwM8x0YONEEWRcw\nSKAJcRoKOaAJDQruf3TwgVRncSu9RtcyiWH7BDSx/Qz8DWhC3KCgQwdm6AIMjAYYQk0MyMYvUlFS\n4NZ03IBQxinyiS/A//9/hAgDx6wC9FhiYH6ApADC5BXAEMLiyOUYihjUkIUACEMze7JpJRoAAAAA\nSUVORK5CYII=\n",
       "prompt_number": 182,
       "text": "\n -a\u22c5r\n\u212f    \n\u2500\u2500\u2500\u2500\u2500\n   2 \n  r  "
      }
     ],
     "prompt_number": 182
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAO8AAABEBAMAAACbhX+nAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/wHnR8hhKrWTbAAAAABJRU5ErkJggg==\n",
       "prompt_number": 183,
       "text": "\n                           -\u03bd     \n       \u239b      ________    \u239e       \n       \u239c     \u2571      2     \u239f       \n -\u03bd  \u03bd \u239c    \u2571      k      \u239f       \na  \u22c5k \u22c5\u239c   \u2571   1 + \u2500\u2500  + 1\u239f  \u22c5\u0393(\u03bd)\n       \u239c  \u2571         2     \u239f       \n       \u239d\u2572\u2571         a      \u23a0       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n             \u0393(\u03bd + 1)             "
      }
     ],
     "prompt_number": 183
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 184
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 185
    },
    {
     "cell_type": "code",
     "input": "r**nu*exp(-a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAANBAMAAAAZPSD6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAAtElEQVQYGWNgAAKeDwy5IBoKlNRN1AUTwBzeB8wTYMIMDFwK/ozcAhAZpgNcAWCZ\nqFWrVgXYO9xj8IQqZG+QQmhhkGbQYRCH8vldIVog3DaGHQGLgExGJSMl62CQWFCUAViKK1RLoBjI\ncrVXSAcLsB9gVgAzYESLJIMDmH2SgQeiBybDsAzK+h4ENhMuzsCwFcJm/oEkBmayfIaIgGgHFEnG\nCVBuIQNrA4oM0wEo1ytUGUUCAHvZHiD635GFAAAAAElFTkSuQmCC\n",
       "prompt_number": 186,
       "text": "\n \u03bd  -a\u22c5r\nr \u22c5\u212f    "
      }
     ],
     "prompt_number": 186
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR4AAABBBAMAAAD1SNZzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 187,
       "text": "\n                         -\u03bd - 3/2           \n                 \u239b 2    \u239e                   \n \u03bd + 1    -\u03bd - 3 \u239ca     \u239f                   \n2     \u22c5a\u22c5k      \u22c5\u239c\u2500\u2500 + 1\u239f        \u22c5\u0393(\u03bd + 3/2)\n                 \u239c 2    \u239f                   \n                 \u239dk     \u23a0                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                     ___                    \n                   \u2572\u2571 \u03c0                     "
      }
     ],
     "prompt_number": 187
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANIAAAASBAMAAADcX53FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 188,
       "text": "\n \u03bd                         \nr \u22c5(-sinh(a\u22c5r) + cosh(a\u22c5r))"
      }
     ],
     "prompt_number": 188
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "input": "r**(nu-1)*exp(-a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAQBAMAAABdIsRgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA0klEQVQoFWNgQAXFqFwsPJnJWATRhHAp4fnAkAtViqJESd1EXTABLMP7gHkCFiVc\nCv6M3AIQJUwHuAJYV61atZSBAWpKFJAXYO9wj8ETqpW9QQrKgikBc6UZdBjEoRL8rgHYlLQx7AhY\nBJRgVDJSsg6GqWj6BlQcFGUA5nOFagkUA1mu9grpMAUQmv0AswKKSIskgwOKAMNJBh6IKXDhZXAW\nlPE9CG4vVGQrmhLmH2gCDCyf0URAAg4oYowTULhATiEDawOKGNMBFC6Q4xWqjCIEAFHkJyRTm6qp\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 189,
       "text": "\n \u03bd - 1  -a\u22c5r\nr     \u22c5\u212f    "
      }
     ],
     "prompt_number": 189
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQMAAABBBAMAAAAnGNwrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3qrx95nkC0VXiH5kio6J22LHkfA/PzaTA3s5QcgAAAAASUVORK5CYII=\n",
       "prompt_number": 190,
       "text": "\n                   -\u03bd - 1/2           \n           \u239b 2    \u239e                   \n \u03bd  -\u03bd - 1 \u239ca     \u239f                   \n2 \u22c5k      \u22c5\u239c\u2500\u2500 + 1\u239f        \u22c5\u0393(\u03bd + 1/2)\n           \u239c 2    \u239f                   \n           \u239dk     \u23a0                   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                  ___                 \n                \u2572\u2571 \u03c0                  "
      }
     ],
     "prompt_number": 190
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAAUBAMAAABv64ynAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nkIsqbZU1Eyc1fWpL6ui2IBdV2nDxD1Vfy+sR1DQuAAAAAElFTkSuQmCC\n",
       "prompt_number": 191,
       "text": "\n \u03bd - 1                         \nr     \u22c5(-sinh(a\u22c5r) + cosh(a\u22c5r))"
      }
     ],
     "prompt_number": 191
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "input": "j = Symbol(\"j\", integer=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 192
    },
    {
     "cell_type": "code",
     "input": "r**(nu-j)*exp(-a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAQBAMAAAClwj+XAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAABGUlEQVQoFWNgQAILgewGJD5OpgNQZgFOWTQJYhQGHQBqAirk+cCQC2RBAZNSsVOh\nGIwHojkufACSCxgYeB8wTwAJQMAilg8CG9mgnKhVq1YFsHA8YFmWti1tAdMBrgCQBCtQdCnzF94H\nLEj6QBI8B4DEAgYG9gYpIAMKGDcwHWAsgPEgNHsDkF7AwMDvCjYQIsiRwN3KpQBkMyoZKUGEGPhB\n9AIGButgIM2quhjEZWBYGrSQ3QBIu9orpIMFGBjugegDUE4NgxGUBaVaJBkcIMwFS5FkeCYwHEbi\ngpjLYPwiBRgLSPOXqoLsQwZbkTlwtn0DnAllsHxGFwHz/Q0YWFAlGCeg8qE8dgGGCFQJpgOofCiP\nZWkQqhsB+nMz8PoPYCwAAAAASUVORK5CYII=\n",
       "prompt_number": 193,
       "text": "\n -j + \u03bd  -a\u22c5r\nr      \u22c5\u212f    "
      }
     ],
     "prompt_number": 193
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAiYAAABFCAMAAABJ792pAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nN3RZVTB6hAAAAABJRU5ErkJggg==\n",
       "prompt_number": 194,
       "text": "\n                \u03bd                      \u239b  j            j       3 \u2502  2  \u2148\u22c5\u03c0\u239e\n j - 2\u22c5\u03bd - 2 \u239bk\u239e                   \u250c\u2500  \u239c- \u2500 + \u03bd + 1, - \u2500 + \u03bd + \u2500 \u2502 k \u22c5\u212f   \u239f\na           \u22c5\u239c\u2500\u239f \u22c5\u0393(-j + 2\u22c5\u03bd + 2)\u22c5 \u251c\u2500  \u239c  2            2       2 \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n             \u239d2\u23a0                  2\u2575 1 \u239c                         \u2502     2  \u239f\n                                       \u239d         \u03bd + 1           \u2502    a   \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  \u0393(\u03bd + 1)                                 "
      }
     ],
     "prompt_number": 194
    },
    {
     "cell_type": "code",
     "input": "#inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [],
     "prompt_number": 195
    },
    {
     "cell_type": "markdown",
     "source": "6"
    },
    {
     "cell_type": "code",
     "input": "r**nu*exp(-a*r**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA3klEQVQoFWNgwAmEluKUYmBgUeA0wC3Nm8DegFuWwYHTAY8sw1F8kowC+GQbeRsg\n0ucZ2C4gFDIpFTsVijH//+8AEbvHYImQZFjE8kFgIxtCQIcB4nPWVatWLWX+wvuAZQJCkqGCcQGC\nx7iB6QBjAYLP8CUUicORwN3KpQARYFQyUmLeagDksKouhggtDVrIDhIAAld7hXQIq4bBCMJAIlsk\nGRzAXJ4JDIeRxKHMZVCav1QVahyymq1Qjn0DsiiUzfIZyvA3YGDBkGecABViF2CIwJBlOgAVYlka\nhGEvAFg8J9N8qZAQAAAAAElFTkSuQmCC\n",
       "prompt_number": 196,
       "text": "\n        2\n \u03bd  -a\u22c5r \nr \u22c5\u212f     "
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAdBAMAAACUHalYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIQt9Jg6AABgAAAAASUVORK5CYII=\n",
       "prompt_number": 197,
       "text": "\n                  2\n                -k \n                \u2500\u2500\u2500\n \u03bd      -\u03bd - 1  4\u22c5a\nk \u22c5(2\u22c5a)      \u22c5\u212f   "
      }
     ],
     "prompt_number": 197
    },
    {
     "cell_type": "code",
     "input": "inverse_hankel_transform(_, k, r, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVHYiEGaru0Tvmd2J\nMs3RRhFbAAAA3klEQVQoFWNgwAmEluKUYmBgUeA0wC3Nm8DegFuWwYHTAY8sw1F8kowC+GQbeRsg\n0ucZ2C4gFDIpFTsVijH//+8AEbvHYImQZFjE8kFgIxtCQIcB4nPWVatWLWX+wvuAZQJCkqGCcQGC\nx7iB6QBjAYLP8CUUicORwN3KpQARYFQyUmLeagDksKouhggtDVrIDhIAAld7hXQIq4bBCMJAIlsk\nGRzAXJ4JDIeRxKHMZVCav1QVahyymq1Qjn0DsiiUzfIZyvA3YGDBkGecABViF2CIwJBlOgAVYlka\nhGEvAFg8J9N8qZAQAAAAAElFTkSuQmCC\n",
       "prompt_number": 198,
       "text": "\n        2\n \u03bd  -a\u22c5r \nr \u22c5\u212f     "
      }
     ],
     "prompt_number": 198
    },
    {
     "cell_type": "markdown",
     "source": "Trigonometric functions"
    },
    {
     "cell_type": "code",
     "input": "a = Symbol(\"a\", positive=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 199
    },
    {
     "cell_type": "markdown",
     "source": "1"
    },
    {
     "cell_type": "code",
     "input": "sin(a*r)/r",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAnBAMAAACs3UZeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABWklEQVQ4Ed2SP0vDQBiHf2lrDkJKooKCKIguIgoKfoCAFB07+AG6CkLiB5BGR10c\nRRwyuRp0cbDksndwc9VFHEQUBcEK9b38OU/knDp5kOS59zneuws/4Odw5dSQJMGMJbLbAqeCsjZS\nAn0XCp6VNlRsVVaLYoUr1naViUDTUwrWEmDMjFpH0Wpv+JGEQ4+dzqHSveoCz0ADZtCO2CcS2qZN\n9gQP2E74JnADrHs1OBFexAsd6hXiCWuLoLUHtFX/WIgPOB5wCdSb2APuqQn2geDcj6XdARKPvQO7\nwlLnCQxxaelUZ4ERTrOesHSqMdge2besczVCPb5o8co1SesQmExTy185fZ33l+k2HLW78a3YbpI1\nXNFAGSxUJr/+JDYU21A4RzOQJdaSWALjJcH+XihrA4T+X2OA+/y3VnnIdbfKQ66zech1Ng+51mYh\n19k85DqbhVwnkYVctV+QsmsDq7FoPAAAAABJRU5ErkJggg==\n",
       "prompt_number": 200,
       "text": "\nsin(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   r    "
      }
     ],
     "prompt_number": 200
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABZsAAAGLCAMAAACWSbzTAAAAeFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 201,
       "text": "\n\u23a7          ___________                                                        \n\u23aa         \u2571    2    2                                                         \n\u23aa        \u2571  - a  + k   \u239b                               \u239b 2          \u239e         \n\u23aaa\u22c5k\u22c5   \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d-\u03bd\u22c5(\u03bd - 2)\u22c5(\u03bd - 1) + \u03bd\u22c5(\u03bd - 1)\u22c5\u239d\u03bd  - 4\u22c5\u03bd + 4\u23a0 + (\u03bd - 2\n\u23aa      \u2571         2                                                            \n\u23aa    \u2572\u2571         k                                                             \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                 \u23a7    \u239b  \u239b           \u239b 2    \u239e         \u239b 2                \u239e   \n\u23aa                 \u23aa    \u239c  \u239c           \u239ca     \u239f       2 \u239ca \u22c5(\u03bd - 2)        \u239f   \n\u23aa                 \u23aa    \u239c  \u239c  (-\u03bd + 2)\u22c5\u239c\u2500\u2500 - 1\u239f      k \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u03bd + 2\u239f   \n\u23aa                 \u23aa    \u239c  \u239c           \u239c 2    \u239f         \u239c     2            \u239f   \n\u23aa                 \u23aa    \u239c  \u239c           \u239dk     \u23a0         \u239d    k             \u23a0   \n\u23aa                 \u23aa    \u239ca\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 +\n\u23aa                 \u23aa    \u239c  \u239c 2  2      2        2    2         \u239b 2          \u239e  \n\u23aa                 \u23aa    \u239c  \u239ca \u22c5\u03bd    4\u22c5a \u22c5\u03bd   3\u22c5a    a \u22c5(\u03bd - 2)\u22c5\u239d\u03bd  - 4\u22c5\u03bd + 3\u23a0  \n\u23aa                 \u23aa    \u239c  \u239c\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500                              \n\u23aa                 \u23aa    \u239c  \u239c   2       2       2                               \n\u23a8                 \u23aa    \u239c  \u239d  k       k       k                                \n\u23aa                 \u23aaa\u22c5\u03bd\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                 \u23aa    \u239c                                                      \n\u23aa                 \u23aa    \u239c                                                      \n\u23aa                 \u23aa    \u239c                                                      \n\u23aa                 \u23aa    \u239c                                                      \n\u23aa                 \u23aa    \u239c                                                     \u2571\n\u23aa             ___ \u23aa    \u239d                                                   \u2572\u2571 \n\u23aa           \u2572\u2571 \u03c0 \u22c5\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23a8                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                         \u2572\u2571\n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                         \u2500\u2500\n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23a9                                                           \n\u23aa           \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                             \n\u23a9                                                                             \n\n                                                                              \n                                                                              \n 2 \u239b 2          \u239e\u239e    \u239b            \u239ba\u239e\u239e   \u239b    \u239b 2    2\u239e        2             \n) \u22c5\u239d\u03bd  - 4\u22c5\u03bd + 3\u23a0\u23a0\u22c5cos\u239c(\u03bd - 2)\u22c5asin\u239c\u2500\u239f\u239f + \u239d- \u03bd\u22c5\u239da  - k \u23a0\u22c5(\u03bd - 2) \u22c5(\u03bd - 1) + \u03bd\u22c5\n                      \u239d            \u239dk\u23a0\u23a0                                       \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                  ___________                                 \n                                 \u2571    2    2                                  \n                          3     \u2571  - a  + k                   \u239b 2          \u239e  \n                         k \u22c5   \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03bd - 2)\u22c5(\u03bd - 1)\u22c5\u239d\u03bd  - 4\u22c5\u03bd + 3\u23a0  \n                              \u2571         2                                     \n                            \u2572\u2571         k                                      \n                                                                              \n                                                       -\u03bd                     \n                                     \u239b       _________\u239e                       \n                                     \u239c      \u2571  2    2 \u239f   \u239b            2      \n                           -\u03bd - 1  \u03bd \u239ca + \u2572\u2571  a  - k  \u239f   \u239c           k       \n                      \u03c0\u22c5a\u22c5a      \u22c5k \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f  \u22c5\u239ca - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                     \u239d       a        \u23a0   \u239c           ________\n                                                          \u239c          \u2571  2    2\n                                                          \u239d    a + \u2572\u2571  a  - k \n                      \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                    \u239b 2    2\u239e  \u239b  \u03bd   1\u239e  \u239b\u03bd   1\u239e             \n                                    \u239da  - k \u23a0\u22c5\u0393\u239c- \u2500 + \u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f             \n                                               \u239d  2   2\u23a0  \u239d2   2\u23a0             \n                                                                              \n    \u239b 2             \u239e\u239e                                                        \n  2 \u239ca \u22c5(-\u03bd + 2)    \u239f\u239f                                                        \n k \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 1\u239f\u239f                                                        \n    \u239c      2        \u239f\u239f      ____    \u239b            \u239b       ____\u239e\u239e               \n    \u239d     k         \u23a0\u239f     \u2571 1      \u239c            \u239c      \u2571 1  \u239f\u239f               \n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5   \u2571  \u2500\u2500 \u22c5sin\u239c(\u03bd - 2)\u22c5asin\u239ca\u22c5   \u2571  \u2500\u2500 \u239f\u239f               \n      2              \u239f   \u2571    2     \u239c            \u239c    \u2571    2 \u239f\u239f               \n     a \u22c5\u03bd\u22c5(\u03bd - 1)    \u239f \u2572\u2571    k      \u239d            \u239d  \u2572\u2571    k  \u23a0\u23a0               \n                     \u239f                                                        \n                     \u239f                                            \u239b           \n                     \u23a0                                            \u239c       1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   __________                                                     \u239c   2       \n  \u2571    2                                                          \u239d  \u03bd  - 4\u22c5\u03bd \n \u2571    a                                                                       \n\u2571   - \u2500\u2500 + 1                                                                  \n       2                                                                      \n      k                                                                       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                            ___                               \n                                          \u2572\u2571 \u03c0 \u22c5k                             \n                                                                              \n                   \u239b                                                          \n                   \u239c                                               \u239b      ____\n                   \u239c                                               \u239c     \u2571    \n                   \u239c                                       -\u03bd    2 \u239c    \u2571     \n                   \u239c                   \u239b      ________    \u239e     k \u22c5\u239c   \u2571   1 -\n                   \u239c\u239b          2     \u239e \u239c     \u2571      2     \u239f        \u239c  \u2571       \n___  -\u03bd - 1  \u03bd + 1 \u239c\u239c         k      \u239f \u239c    \u2571      k      \u239f        \u239d\u2572\u2571        \n \u03c0 \u22c5a      \u22c5k     \u22c5\u239c\u239c1 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u239c   \u2571   1 - \u2500\u2500  + 1\u239f   + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   \u239c\u239c       \u239b      2\u239e\u239f \u239c  \u2571         2     \u239f                 \u239b \n                   \u239c\u239c     2 \u239c     k \u239f\u239f \u239d\u2572\u2571         a      \u23a0               2 \u239c \n                   \u239c\u239c    a \u22c5\u239c-1 + \u2500\u2500\u239f\u239f                                   a \u22c5\u239c-\n                   \u239c\u239c       \u239c      2\u239f\u239f                                      \u239c \n                   \u239d\u239d       \u239d     a \u23a0\u23a0                                      \u239d \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                     \u239b  \u03bd   1\u239e  \u239b\u03bd   1\u239e                       \n                                    \u0393\u239c- \u2500 + \u2500\u239f\u22c5\u0393\u239c\u2500 + \u2500\u239f                       \n                                     \u239d  2   2\u23a0  \u239d2   2\u23a0                       \n                                                                              \n                                       \u239b  \u03bd   1  \u03bd   1 \u2502  2\u239e                  \n                               \u256d\u2500\u256e1, 1 \u239c- \u2500 + \u2500  \u2500 + \u2500 \u2502 a \u239f                  \n                               \u2502\u2576\u2510     \u239c  2   2  2   2 \u2502 \u2500\u2500\u239f                  \n                               \u2570\u2500\u256f2, 2 \u239c               \u2502  2\u239f                  \n                                       \u239d  1/2      0   \u2502 k \u23a0                  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                  k                           \n                                                                              \n\n                                                                              \n                                                                              \n        \u239b 2            2        2\u239e           \u239b   2            2\u239e \u239b 2          \n(\u03bd - 1)\u22c5\u239da \u22c5(\u03bd - 2) - k \u22c5\u03bd + 2\u22c5k \u23a0 + (\u03bd - 2)\u22c5\u239d- a \u22c5(\u03bd - 2) - k \u23a0\u22c5\u239d\u03bd  - 4\u22c5\u03bd + 3\n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n \u239e                                                                            \n \u239f                                                                            \n\u2500\u239f                                                                            \n_\u239f                                                                            \n \u239f                                                                            \n \u23a0                                                                            \n\u2500\u2500                                                                            \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                        \u239e     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n            2                           \u239e    \u239b            \u239b       ____\u239e\u239e\u239f     \n           \u03bd  - 4\u22c5\u03bd + 4          \u03bd - 2  \u239f    \u239c            \u239c      \u2571 1  \u239f\u239f\u239f     \n\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5cos\u239c(\u03bd - 2)\u22c5asin\u239ca\u22c5   \u2571  \u2500\u2500 \u239f\u239f\u239f     \n              \u239b 2          \u239e   \u03bd\u22c5(\u03bd - 1)\u239f    \u239c            \u239c    \u2571    2 \u239f\u239f\u239f     \n+ 3   (\u03bd - 2)\u22c5\u239d\u03bd  - 4\u22c5\u03bd + 3\u23a0            \u23a0    \u239d            \u239d  \u2572\u2571    k  \u23a0\u23a0\u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u239f     \n                                                                        \u23a0     \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for\n                                                                              \n                                                                              \n                                                                              \n         -\u03bd - 1\u239e                                                              \n____    \u239e      \u239f                                                              \n  2     \u239f      \u239f                                                              \n k      \u239f      \u239f                                                              \n \u2500\u2500  + 1\u239f      \u239f                                                              \n  2     \u239f      \u239f                                                              \n a      \u23a0      \u239f                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                              \n     2\u239e        \u239f                                                              \n    k \u239f        \u239f                                                              \n1 + \u2500\u2500\u239f        \u239f                                                              \n     2\u239f        \u239f                                                              \n    a \u23a0        \u23a0                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                            fo\n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                                              \n                                                                              \n\n                                        \n                                        \n\u239e\u239e    \u239b            \u239ba\u239e\u239e                 \n\u23a0\u23a0\u22c5sin\u239c(\u03bd - 2)\u22c5asin\u239c\u2500\u239f\u239f                 \n      \u239d            \u239dk\u23a0\u23a0                 \n                              2 \u25021 \u2502    \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for a \u22c5\u2502\u2500\u2500\u2502 < 1\n                                \u2502 2\u2502    \n                                \u2502k \u2502    \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                              \u2502 2\u2502      \n                              \u2502k \u2502      \n                          for \u2500\u2500\u2500\u2500 < 1  \n                                2       \n                               a        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n  2 \u25021 \u2502                                \n a \u22c5\u2502\u2500\u2500\u2502 < 1                            \n    \u2502 2\u2502                                \n    \u2502k \u2502                                \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n  \u2502 2\u2502                                  \n  \u2502k \u2502                                  \nr \u2500\u2500\u2500\u2500 < 1                              \n    2                                   \n   a                                    \n                                        \n                                        \n                                        \n                                        \notherwise                               \n                                        \n                                        \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                otherwise   \n                                        \n                                        "
      }
     ],
     "prompt_number": 201
    },
    {
     "cell_type": "markdown",
     "source": "2"
    },
    {
     "cell_type": "code",
     "input": "sin(a*r)/r**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAAnBAMAAACs3UZeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nd86Lr4B9AAAAAElFTkSuQmCC\n",
       "prompt_number": 202,
       "text": "\nsin(a\u22c5r)\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    2   \n   r    "
      }
     ],
     "prompt_number": 202
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABDcAAAFhCAMAAABTfLGoAAAAbFBMVEX///8AAAAAAAAAAAAAAAAA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\nAalids5ZI1KWAAAAAElFTkSuQmCC\n",
       "prompt_number": 203,
       "text": "\n\u23a7                                ___________                                  \n\u23aa                               \u2571    2    2                                   \n\u23aa                              \u2571  - a  + k   \u239b   2          \u239e    \u239b            \n\u23aa                    - a\u22c5k\u22c5   \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5\u239d- \u03bd  + 3\u22c5\u03bd - 2\u23a0\u22c5cos\u239c(\u03bd - 1)\u22c5asin\n\u23aa                            \u2571         2                         \u239d            \n\u23aa                          \u2572\u2571         k                                       \n\u23aa                    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                             \n\u23aa                                                                            \u2571\n\u23aa                                                                   2       \u2571 \n\u23aa                                                                  k \u22c5\u03bd\u22c5   \u2571  \n\u23aa                                                                         \u2571   \n\u23aa                                                                       \u2572\u2571    \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                          -\u03bd \n\u23aa                                                                       \u03c0\u22c5a  \u22c5\n\u23aa                                                                             \n\u23aa                                                                       \u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                             \n\u23aa                                                                           2\u22c5\n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa      \u23a7    \u239b  \u239b         \u239b 2    \u239e      \u239b 2                \u239e\u239e                  \n\u23aa      \u23aa    \u239c  \u239c         \u239ca     \u239f    2 \u239ca \u22c5(\u03bd - 1)        \u239f\u239f                  \n\u23aa      \u23aa    \u239c  \u239c(-\u03bd + 1)\u22c5\u239c\u2500\u2500 - 1\u239f   k \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u03bd + 1\u239f\u239f                  \n\u23aa      \u23aa    \u239c  \u239c         \u239c 2    \u239f      \u239c     2            \u239f\u239f      ____    \u239b   \n\u23aa      \u23aa    \u239c  \u239c         \u239dk     \u23a0      \u239d    k             \u23a0\u239f     \u2571 1      \u239c   \n\u23aa      \u23aa    \u239ca\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5   \u2571  \u2500\u2500 \u22c5sin\u239c(\u03bd \n\u23aa      \u23aa    \u239c  \u239c   2  2      2        2         \u239b 2      \u239e \u239f   \u2571    2     \u239c   \n\u23aa      \u23aa    \u239c  \u239c  a \u22c5\u03bd    2\u22c5a \u22c5\u03bd     a \u22c5(\u03bd - 1)\u22c5\u239d\u03bd  - 2\u22c5\u03bd\u23a0 \u239f \u2572\u2571    k      \u239d   \n\u23a8      \u23aa    \u239c  \u239c  \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500                           \u239f                  \n\u23aa      \u23aa    \u239c  \u239c     2       2                             \u239f                  \n\u23aa      \u23aa    \u239c  \u239d    k       k                              \u23a0                  \n\u23aa      \u23aa2\u22c5a\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u23aa    \u239c                                          __________             \n\u23aa      \u23aa    \u239c                                         \u2571    2                  \n\u23aa      \u23aa    \u239c                                        \u2571    a                   \n\u23aa      \u23aa    \u239c                                       \u2571   - \u2500\u2500 + 1              \n\u23aa      \u23aa    \u239c                                      \u2571       2                  \n\u23aa  ___ \u23aa    \u239d                                    \u2572\u2571       k                   \n\u23aa\u2572\u2571 \u03c0 \u22c5\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u23a8                                                                      \n\u23aa      \u23aa                                                                      \n\u23aa      \u23aa                                                                      \n\u23aa      \u23aa                                                                      \n\u23aa      \u23aa                                                                     \u239b\n\u23aa      \u23aa                                                                     \u239c\n\u23aa      \u23aa                                                          ___  -\u03bd  \u03bd \u239c\n\u23aa      \u23aa                                                        \u2572\u2571 \u03c0 \u22c5a  \u22c5k \u22c5\u239c\n\u23aa      \u23aa                                                                     \u239c\n\u23aa      \u23aa                                                                     \u239d\n\u23aa      \u23aa                                                        \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa      \u23aa                                                                 \u239b  \u03bd \n\u23aa      \u23aa                                                                \u0393\u239c- \u2500 \n\u23aa      \u23aa                                                                 \u239d  2 \n\u23aa      \u23aa                                                                      \n\u23aa      \u23aa                                                                   \u239b  \n\u23aa      \u23aa                                                           \u256d\u2500\u256e1, 1 \u239c- \n\u23aa      \u23aa                                                           \u2502\u2576\u2510     \u239c  \n\u23aa      \u23aa                                                           \u2570\u2500\u256f2, 2 \u239c  \n\u23aa      \u23a9                                                                   \u239d  \n\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                                             \n\u23a9                                                                             \n\n                                                                              \n                                                                              \n\u239ba\u239e\u239e   \u239b 2            2      2   \u239b 2    2\u239e        2\u239e    \u239b            \u239ba\u239e\u239e     \n\u239c\u2500\u239f\u239f + \u239da \u22c5(\u03bd - 1) - k \u22c5\u03bd + k  - \u239da  - k \u23a0\u22c5(\u03bd - 1) \u23a0\u22c5sin\u239c(\u03bd - 1)\u22c5asin\u239c\u2500\u239f\u239f     \n\u239dk\u23a0\u23a0                                                    \u239d            \u239dk\u23a0\u23a0     \n                                                                              \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     \n___________                                                                   \n    2    2                                                                    \n - a  + k                                                                     \n \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u22c5(\u03bd - 2)\u22c5(\u03bd - 1)                                                   \n      2                                                                       \n     k                                                                        \n                                                                              \n                     -\u03bd                                                       \n   \u239b       _________\u239e                                                         \n   \u239c      \u2571  2    2 \u239f                                                         \n \u03bd \u239ca + \u2572\u2571  a  - k  \u239f                                                         \nk \u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f                                                         \n   \u239d       a        \u23a0                                                         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                       \n \u239b  \u03bd    \u239e  \u239b\u03bd    \u239e                                                           \n\u0393\u239c- \u2500 + 1\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                                                           \n \u239d  2    \u23a0  \u239d2    \u23a0                                                           \n                                                                              \n                                                                            \u239e \n                                                                            \u239f \n                                                                            \u239f \n         \u239b       ____\u239e\u239e                                                     \u239f \n         \u239c      \u2571 1  \u239f\u239f                                                     \u239f \n- 1)\u22c5asin\u239ca\u22c5   \u2571  \u2500\u2500 \u239f\u239f                                                     \u239f \n         \u239c    \u2571    2 \u239f\u239f   \u239b      2          \u239e    \u239b            \u239b       ____\u239e\u239e\u239f \n         \u239d  \u2572\u2571    k  \u23a0\u23a0   \u239c     \u03bd  - 2\u22c5\u03bd + 1\u239f    \u239c            \u239c      \u2571 1  \u239f\u239f\u239f \n                          \u239c-1 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5cos\u239c(\u03bd - 1)\u22c5asin\u239ca\u22c5   \u2571  \u2500\u2500 \u239f\u239f\u239f \n                          \u239d        \u03bd - 1    \u23a0    \u239c            \u239c    \u2571    2 \u239f\u239f\u239f \n                                                 \u239d            \u239d  \u2572\u2571    k  \u23a0\u23a0\u239f \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f \n                                                2                           \u239f \n                                               \u03bd  - 2\u22c5\u03bd                     \u239f \n                                                                            \u239f \n                                                                            \u239f \n                                                                            \u239f \n                                                                            \u23a0 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n  ___                                                                         \n\u2572\u2571 \u03c0 \u22c5k                                                                       \n                                                                              \n                   -\u03bd                                                         \n      ________    \u239e                                                           \n     \u2571      2     \u239f                                                           \n    \u2571      k      \u239f                                                           \n   \u2571   1 - \u2500\u2500  + 1\u239f                                                           \n  \u2571         2     \u239f                                                           \n\u2572\u2571         a      \u23a0                                                           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                         \n   \u239e  \u239b\u03bd    \u239e                                                                 \n+ 1\u239f\u22c5\u0393\u239c\u2500 + 1\u239f                                                                 \n   \u23a0  \u239d2    \u23a0                                                                 \n                                                                              \n\u03bd      \u03bd     \u2502  2\u239e                                                            \n\u2500 + 1  \u2500 + 1 \u2502 a \u239f                                                            \n2      2     \u2502 \u2500\u2500\u239f                                                            \n             \u2502  2\u239f                                                            \n1/2      0   \u2502 k \u23a0                                                            \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n        2                                                                     \n                                                                              \n\n                                 \n                                 \n                                 \n                                 \n                                 \n                       2 \u25021 \u2502    \n                  for a \u22c5\u2502\u2500\u2500\u2502 < 1\n                         \u2502 2\u2502    \n                         \u2502k \u2502    \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                       \u2502 2\u2502      \n                       \u2502k \u2502      \n                   for \u2500\u2500\u2500\u2500 < 1  \n                         2       \n                        a        \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n      2 \u25021 \u2502                     \n for a \u22c5\u2502\u2500\u2500\u2502 < 1                 \n        \u2502 2\u2502                     \n        \u2502k \u2502                     \n                                 \n                                 \n                                 \n                                 \n                                 \n                                 \n      \u2502 2\u2502                       \n      \u2502k \u2502                       \n  for \u2500\u2500\u2500\u2500 < 1                   \n        2                        \n       a                         \n                                 \n                                 \n                                 \n                                 \n    otherwise                    \n                                 \n                                 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500     otherwise   \n                                 \n                                 "
      }
     ],
     "prompt_number": 203
    },
    {
     "cell_type": "markdown",
     "source": "3"
    },
    {
     "cell_type": "code",
     "input": "r**nu*sin(a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAASBAMAAAAHyPWiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 204,
       "text": "\n \u03bd         \nr \u22c5sin(a\u22c5r)"
      }
     ],
     "prompt_number": 204
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAs0AAADxCAMAAAAHtnpMAAAASFBMVEX///8AAAAAAAAAAAAAAAAA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\npE0rC7SRk4+hwJa4OKbHuZVMAZMC/w+JKMuB2U516gAAAABJRU5ErkJggg==\n",
       "prompt_number": 205,
       "text": "\n\u23a7                                                   -\u03bd - 3/2                  \n\u23aa                                         \u239b   2    \u239e                          \n\u23aa                         \u03bd + 1    -\u03bd - 3 \u239c  a     \u239f          \u2148\u22c5\u03c0             \n\u23aa                        2     \u22c5a\u22c5k      \u22c5\u239c- \u2500\u2500 + 1\u239f        \u22c5\u212f   \u22c5\u0393(\u03bd + 3/2)  \n\u23aa                                         \u239c   2    \u239f                          \n\u23aa                                         \u239d  k     \u23a0                          \n\u23aa                        \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \n\u23aa                                                 ___                         \n\u23aa                                               \u2572\u2571 \u03c0                          \n\u23aa                                                                             \n\u23aa                                               -\u03bd - 3/2                      \n\u23aa                                       \u239b     2\u239e                              \n\u23aa                    \u03bd + 1  -2\u22c5\u03bd - 2  \u03bd \u239c    k \u239f                              \n\u23aa                   2     \u22c5a        \u22c5k \u22c5\u239c1 - \u2500\u2500\u239f        \u22c5sin(\u03c0\u22c5(\u03bd + 1))\u22c5\u0393(\u03bd + \n\u23aa                                       \u239c     2\u239f                              \n\u23aa                                       \u239d    a \u23a0                              \n\u23aa                   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                                                 ___                         \n\u23aa                                               \u2572\u2571 \u03c0                          \n\u23aa                                                                             \n\u23aa                     \u23a7                           -\u03bd - 3/2                    \n\u23aa                     \u23aa                 \u239b   2    \u239e                            \n\u23a8                     \u23aa                 \u239c  a     \u239f                            \n\u23aa                     \u23aa              -a\u22c5\u239c- \u2500\u2500 + 1\u239f        \u22c5\u0393(\u03bd + 3/2)         \n\u23aa                     \u23aa                 \u239c   2    \u239f                            \n\u23aa \u03bd + 1   ___  -\u03bd - 2 \u23aa                 \u239d  k     \u23a0                            \n\u23aa2     \u22c5\u2572\u2571 \u03c0 \u22c5k      \u22c5\u23aa              \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         \n\u23aa                     \u23aa                            \u03c0\u22c5k                        \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa                           -\u03bd - 3/2                    \n\u23aa                     \u23aa                   \u239b     2\u239e                            \n\u23aa                     \u23a8 -2\u22c5\u03bd - 2  2\u22c5\u03bd + 2 \u239c    k \u239f                            \n\u23aa                     \u23aaa        \u22c5k       \u22c5\u239c1 - \u2500\u2500\u239f        \u22c5sin(\u03c0\u22c5(\u03bd + 1))\u22c5\u0393(\u03bd \n\u23aa                     \u23aa                   \u239c     2\u239f                            \n\u23aa                     \u23aa                   \u239d    a \u23a0                            \n\u23aa                     \u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n\u23aa                     \u23aa                              \u03c0                        \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa                            \u239b       \u2502  2\u239e              \n\u23aa                     \u23aa                    \u256d\u2500\u256e1, 1 \u239c-\u03bd   0 \u2502 a \u239f              \n\u23aa                     \u23aa                    \u2502\u2576\u2510     \u239c       \u2502 \u2500\u2500\u239f              \n\u23aa                     \u23aa                    \u2570\u2500\u256f2, 2 \u239c1/2  0 \u2502  2\u239f              \n\u23a9                     \u23a9                            \u239d       \u2502 k \u23a0              \n\n                                        \n                                        \n                                        \n                                        \n                                        \n                              2 \u25021 \u2502    \n                         for a \u22c5\u2502\u2500\u2500\u2502 < 1\n                                \u2502 2\u2502    \n                                \u2502k \u2502    \n                                        \n                                        \n                                        \n                                        \n3/2)                                    \n                              \u2502 2\u2502      \n                              \u2502k \u2502      \n\u2500\u2500\u2500\u2500                      for \u2500\u2500\u2500\u2500 < 1  \n                                2       \n                               a        \n                                        \n                                        \n                                        \n                                        \n                                        \n                                        \n             2 \u25021 \u2502                     \n        for a \u22c5\u2502\u2500\u2500\u2502 < 1     otherwise   \n               \u2502 2\u2502                     \n               \u2502k \u2502                     \n                                        \n                                        \n                                        \n                                        \n+ 3/2)                                  \n             \u2502 2\u2502                       \n             \u2502k \u2502                       \n\u2500\u2500\u2500\u2500\u2500\u2500   for \u2500\u2500\u2500\u2500 < 1                   \n               2                        \n              a                         \n                                        \n                                        \n                                        \n           otherwise                    \n                                        \n                                        "
      }
     ],
     "prompt_number": 205
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "input": "r**(nu-1)*sin(a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAUBAMAAACABSXcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nc+HgmZi3JyRn82zatkdiAj/EgFYwyQe++wAAAABJRU5ErkJggg==\n",
       "prompt_number": 206,
       "text": "\n \u03bd - 1         \nr     \u22c5sin(a\u22c5r)"
      }
     ],
     "prompt_number": 206
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAjkAAADTCAMAAABUdfKSAAAAQlBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 207,
       "text": "\n\u23a7                                                                             \n\u23aa                                  0                                     for a\n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                              -\u03bd - 1/2                       \n\u23aa                                      \u239b     2\u239e                               \n\u23aa                 \u03bd   ___  -2\u22c5\u03bd - 1  \u03bd \u239c    k \u239f                               \n\u23aa                2 \u22c5\u2572\u2571 \u03c0 \u22c5a        \u22c5k \u22c5\u239c1 - \u2500\u2500\u239f                               \n\u23aa                                      \u239c     2\u239f                               \n\u23aa                                      \u239d    a \u23a0                               \n\u23aa                \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                   for \n\u23aa                             \u0393(-\u03bd + 1/2)                                     \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa \u03bd   ___  -\u03bd - 1 \u23a7                                          2 \u25021 \u2502           \n\u23aa2 \u22c5\u2572\u2571 \u03c0 \u22c5k      \u22c5\u23aa                 0                   for a \u22c5\u2502\u2500\u2500\u2502 < 1     ot\n\u23aa                 \u23aa                                            \u2502 2\u2502           \n\u23a8                 \u23aa                                            \u2502k \u2502           \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa                           -\u03bd - 1/2                        \n\u23aa                 \u23aa                   \u239b     2\u239e                                \n\u23aa                 \u23aa -2\u22c5\u03bd - 1  2\u22c5\u03bd + 1 \u239c    k \u239f                                \n\u23aa                 \u23aaa        \u22c5k       \u22c5\u239c1 - \u2500\u2500\u239f                                \n\u23aa                 \u23aa                   \u239c     2\u239f               \u2502 2\u2502             \n\u23aa                 \u23a8                   \u239d    a \u23a0               \u2502k \u2502             \n\u23aa                 \u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   for \u2500\u2500\u2500\u2500 < 1         \n\u23aa                 \u23aa            \u0393(-\u03bd + 1/2)                     2              \n\u23aa                 \u23aa                                           a               \n\u23aa                 \u23aa                                                           \n\u23aa                 \u23aa           \u239b              \u2502  2\u239e                            \n\u23aa                 \u23aa   \u256d\u2500\u256e1, 1 \u239c-\u03bd + 1/2  1/2 \u2502 a \u239f                            \n\u23aa                 \u23aa   \u2502\u2576\u2510     \u239c              \u2502 \u2500\u2500\u239f         otherwise          \n\u23aa                 \u23aa   \u2570\u2500\u256f2, 2 \u239c  1/2      0  \u2502  2\u239f                            \n\u23aa                 \u23aa           \u239d              \u2502 k \u23a0                            \n\u23aa                 \u23a9                                                           \n\u23a9                                                                             \n\n2 \u25021 \u2502    \n \u22c5\u2502\u2500\u2500\u2502 < 1\n  \u2502 2\u2502    \n  \u2502k \u2502    \n          \n          \n          \n          \n          \n\u2502 2\u2502      \n\u2502k \u2502      \n\u2500\u2500\u2500\u2500 < 1  \n  2       \n a        \n          \n          \nherwise   \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          \n          "
      }
     ],
     "prompt_number": 207
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "input": "r**nu*cos(a*r)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAASBAMAAAA+r0T5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 208,
       "text": "\n \u03bd         \nr \u22c5cos(a\u22c5r)"
      }
     ],
     "prompt_number": 208
    },
    {
     "cell_type": "code",
     "input": "hankel_transform(_, r, k, nu)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAksAAADTCAMAAACmyac3AAAARVBMVEX///8AAAAAAAAAAAAAAAAA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\nbslaDlT7YmltcWubn/NlDmQO+Djwf+sEOrpWAzoxAAAAAElFTkSuQmCC\n",
       "prompt_number": 209,
       "text": "\n\u23a7                                                                             \n\u23aa                                    0                                       f\n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa                                                  -\u03bd - 3/2                   \n\u23aa                                          \u239b     2\u239e                           \n\u23aa                 \u03bd + 1   ___  -2\u22c5\u03bd - 2  \u03bd \u239c    k \u239f                           \n\u23aa                2     \u22c5\u2572\u2571 \u03c0 \u22c5a        \u22c5k \u22c5\u239c1 - \u2500\u2500\u239f                           \n\u23aa                                          \u239c     2\u239f                           \n\u23aa                                          \u239d    a \u23a0                           \n\u23aa                \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                   \n\u23aa                               \u0393(-\u03bd - 1/2)                                   \n\u23aa                                                                             \n\u23aa                                                                             \n\u23aa \u03bd + 1   ___  -\u03bd - 2 \u23a7                                          2 \u25021 \u2502       \n\u23aa2     \u22c5\u2572\u2571 \u03c0 \u22c5k      \u22c5\u23aa                 0                   for a \u22c5\u2502\u2500\u2500\u2502 < 1   \n\u23aa                     \u23aa                                            \u2502 2\u2502       \n\u23a8                     \u23aa                                            \u2502k \u2502       \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa                           -\u03bd - 3/2                    \n\u23aa                     \u23aa                   \u239b     2\u239e                            \n\u23aa                     \u23aa -2\u22c5\u03bd - 2  2\u22c5\u03bd + 2 \u239c    k \u239f                            \n\u23aa                     \u23aaa        \u22c5k       \u22c5\u239c1 - \u2500\u2500\u239f                            \n\u23aa                     \u23aa                   \u239c     2\u239f               \u2502 2\u2502         \n\u23aa                     \u23a8                   \u239d    a \u23a0               \u2502k \u2502         \n\u23aa                     \u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   for \u2500\u2500\u2500\u2500 < 1     \n\u23aa                     \u23aa            \u0393(-\u03bd - 1/2)                     2          \n\u23aa                     \u23aa                                           a           \n\u23aa                     \u23aa                                                       \n\u23aa                     \u23aa              \u239b        \u2502  2\u239e                           \n\u23aa                     \u23aa      \u256d\u2500\u256e1, 1 \u239c-\u03bd   0  \u2502 a \u239f                           \n\u23aa                     \u23aa      \u2502\u2576\u2510     \u239c        \u2502 \u2500\u2500\u239f            otherwise      \n\u23aa                     \u23aa      \u2570\u2500\u256f2, 2 \u239c0   1/2 \u2502  2\u239f                           \n\u23aa                     \u23aa              \u239d        \u2502 k \u23a0                           \n\u23aa                     \u23a9                                                       \n\u23a9                                                                             \n\n    2 \u25021 \u2502    \nor a \u22c5\u2502\u2500\u2500\u2502 < 1\n      \u2502 2\u2502    \n      \u2502k \u2502    \n              \n              \n              \n              \n              \n    \u2502 2\u2502      \n    \u2502k \u2502      \nfor \u2500\u2500\u2500\u2500 < 1  \n      2       \n     a        \n              \n              \n  otherwise   \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              \n              "
      }
     ],
     "prompt_number": 209
    },
    {
     "cell_type": "code",
     "input": "",
     "language": "python",
     "outputs": []
    }
   ]
  }
 ]
}