{
 "metadata": {
  "name": "",
  "signature": "sha256:d32bc6df33720bfc192b52875bf6f38d17960d836ecb25fdae346161aba38b80"
 },
 "nbformat": 3,
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from sympy import *\n",
      "import mpmath as mp\n",
      "from IPython.display import display, Math, Latex\n",
      "#init_printing(use_latex='mathjax')\n",
      "init_printing()\n",
      "def latex_print( str ):\n",
      "   \"Helper to print LaTeX strings, takes raw string (like r'string')\"    \n",
      "   display(Latex(str))\n",
      "   return"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using sympy notation: for subscript it is a single underscore \\_ and for superscripts they use double underscore  \\_\\_ \n",
      "\n",
      "Define symbols for scattering coefficients eq. (A2)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "var('z, zp1, n, tmp1, tmp2')\n",
      "D__l_n, D__1_n = symbols('D__(l)_n, D__(1)_n', cls=Function);\n",
      "#psi__dash_n = symbols(\"psi__'_n\", cls=Function);\n",
      "#zeta__dash_n = symbols(\"zeta__'_n\", cls=Function);\n",
      "#psi_n = symbols(\"psi_n\", cls=Function);\n",
      "#zeta_n = symbols(\"zeta_n\", cls=Function);\n",
      "psi__dash_n = symbols(\"psi__'_n\");\n",
      "zeta__dash_n = symbols(\"zeta__'_n\");\n",
      "psi_n = symbols(\"psi_n\");\n",
      "zeta_n = symbols(\"zeta_n\");\n",
      "a__l_n, b__l_n = symbols('a__(l)_n, b__(l)_n');\n",
      "a__1_n, b__1_n = symbols('a__(1)_n, b__(1)_n');\n",
      "c__l_n, d__l_n = symbols('c__(l)_n, d__(l)_n');\n",
      "a__lp1_n, b__lp1_n = symbols('a__(l+1)_n, b__(l+1)_n');\n",
      "c__lp1_n, d__lp1_n = symbols('c__(l+1)_n, d__(l+1)_n');\n",
      "a__lm1_n, b__lm1_n = symbols('a__(l-1)_n, b__(l-1)_n');\n",
      "c__lm1_n, d__lm1_n = symbols('c__(l-1)_n, d__(l-1)_n');\n",
      "x_l, m_l, m_lp1 = symbols('x_l, m_l, m_l+1')\n",
      "D__1_n, D__3_n, R_n = symbols('D__(1)_n, D__(3)_n, R_n', cls=Function)\n",
      "#D__1_n(z) = psi__dash_n(z)/psi_n(z)\n",
      "#z*mp.besselj(n,z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "eq. (A2)\n",
      "\n",
      "__WARNING!!__\n",
      "\n",
      "It seems to be two misprints in Yang paper\n",
      "- first sub-equation uses $m_1$ instead of $m_l$\n",
      "- last sub-equation uses $a_n^(l-1)$ instead of $a_n^(l+1)$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqA2_1 = (    d__lp1_n * m_l * psi__dash_n (m_lp1*x_l) \n",
      "            - a__lp1_n * m_l * zeta__dash_n(m_lp1*x_l)\n",
      "            - d__l_n * m_lp1 * psi__dash_n (m_l*x_l) \n",
      "            + a__l_n * m_lp1 * zeta__dash_n(m_l*x_l)\n",
      "            )\n",
      "eqA2_2 = (    c__lp1_n * m_l * psi_n( m_lp1*x_l)\n",
      "            - b__lp1_n * m_l * zeta_n(m_lp1*x_l)\n",
      "            - c__l_n * m_lp1 * psi_n( m_l*x_l) \n",
      "            + b__l_n * m_lp1 * zeta_n(m_l*x_l)\n",
      "            )\n",
      "eqA2_3 = (    c__lp1_n * psi__dash_n (m_lp1*x_l) \n",
      "            - b__lp1_n * zeta__dash_n(m_lp1*x_l)\n",
      "            - c__l_n * psi__dash_n (m_l*x_l) \n",
      "            + b__l_n * zeta__dash_n(m_l*x_l)\n",
      "            )\n",
      "eqA2_4 = (    d__lp1_n * psi_n( m_lp1*x_l)\n",
      "            - a__lp1_n * zeta_n(m_lp1*x_l)\n",
      "            - d__l_n * psi_n( m_l*x_l) \n",
      "            + a__l_n * zeta_n(m_l*x_l)\n",
      "            )\n",
      "print(\"eqA2.1\")\n",
      "display(eqA2_1)\n",
      "print(\"\\neqA2.2\")\n",
      "display(eqA2_2)\n",
      "print(\"\\neqA2.3\")\n",
      "display(eqA2_3)\n",
      "print(\"\\neqA2.4\")\n",
      "display(eqA2_4)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "eqA2.1\n"
       ]
      },
      {
       "latex": [
        "$$a^{(l)}_{n} m_{l+1} \\zeta^{'}_{n}{\\left (m_{l} x_{l} \\right )} - a^{(l+1)}_{n} m_{l} \\zeta^{'}_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l)}_{n} m_{l+1} \\psi^{'}_{n}{\\left (m_{l} x_{l} \\right )} + d^{(l+1)}_{n} m_{l} \\psi^{'}_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAs4AAAAaBAMAAABV8scPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIF0lEQVRoBbVZa4hkxRX+erp77p1+zLZLQNTI\njjgGkRAamUVB0caduKImmUAixETtRFEDUa+7i6I/dhpXFlwRR4wQVMIVNhhiiBOiRogk7YP9EUXb\nYCImwjZR8d20L2ZBFjynHrce0/fRsB7oqlNV53z11bl1bz0akFJqAU2lp2b7UltSGo5z6wduUZWm\nRjUojJ/LGpi6h0K8EbYNk6Labty+UV3Psf5jTrvfHLSDxyNdeTnwU607+bSoxlng57IGpu2hGG/U\n+oZKUe1RhF1sz7YONrLbN7XWIuznyh79fvYtoLzCJU+mRjX+Ej+HNTB1D4V4A6vJHDKUcrSwg3If\nt2db1dvZ7Zta7wTO5cqYk//QgIXCBUumRjW+Ej+HNTB1D4V4A38xTIpqtRbm11GbNN8MBD3m6eQR\n4EP2iDmhOON3rHgyNarxl/g5rIGpeyjEG3jUMCmqzQJbWqh0MuxbesVJ+yA+ucm5C3zClTEnHOdl\nVmwpJevY5G+3bZuGn8kaNHXUSnkseYNxB5JdGq7hXidjKVcCPyFCC7o8IW/ruhO14uUzCZpqqA5R\nXWM95oTj/C9WbCnrQinWWmru4R94WOFnskZCGwV5J7hZvFEA1xrIaVo/AFwMhLEuT8gXL1GV705o\nFFVDr4HgKjHXiYTjTN24MrO8LivmI7dhUmnoVh5V+JmsrXgU5a1xs3jbcU7FNXTntXoHcBCgCZIq\nu174nmyrmBniGe/wyuECah3MjEa/HdFnmuNMr40jwbeXB7LC93XMVMG1CTckfiZrlNRzBPJ436x6\n0bhZvFEI14wh6fty4IEVEPE0Cd7TLXMtrfn5bV5FNcbcj7ku5oTjTI/Tkb09XbxPKxm5i1/uViV+\nBmvaayRs83i/pnpOcDN4F8M1QwnWlH4LMO5kbYBoN6Jkv1Y25fMDtyroYkuFq2JO3qCfvxMSuxFu\nDLuc5oiLP99X+JnbtmQBkDv5iT1IXB3nBDeDNx0FEqT0eCQmwMf0Bpz63eswG+H5HmZ6CM88bnkx\nXPq9ZSNUsQ/WmmMULJ9y55JoKfdFJpKtS5dFuA6Ns7kU0++bNy0Cp7v4xoG1fad844LOvot7ZGxL\nGv5VA4Wfznr3CxSPCxUYjSCTt4zz1qUEN513Jq7hbpjzMeKuVm2Mipiu9Lku1w5i9SIx+YyDc6R6\nzDPajac6Nwrb+jBxCe/D48Alqhzr+qGLbw5UNeq//YdO8HmvuaaNVZ6KHyn8dNbtwzgZP4gkUB5v\nEWfmrXHVfGZvj3cmrqJNmWGOvyN4CI1YQf4PuGJ2iG1t/APuzYSZebjBMzoL50ev0u0IbQu7SR+3\nreOfdEBQ5b7Km+suvnlJZgeoDm5E6TPUxwJLelRjIA3/ZY0/mTW5lgaH8FfMSahc3iLOxDvBBfrK\n1eOdjSt9qrHNnBanxgZmO8Dd1BwsANFqD4cjnAH3ZoLioOU3nlGED6gp7NJ2ZYy917K8g1ciXEOV\nNEtt2eW50iNTMtdDEBxBcwMzsfxYh9RSfa5LHin4Vyv8zay1a4D7aVvXUF1k8iYbEWfirXGVG2ce\n72xc6O4t5rgKczFWV2gu0q8SEeZh4BDwa+9mYgs1KyG+ntHnVMMTnuKs5UEEX5D+b12W+U7KbHw2\nkUJxRnOMRhdzHYEl3wX5hkzGP0KejL+ZtXiN2DX8hOK8h2xYMnmfNxr9cjT6CMRb40ovkfq8s3GT\n7gHNnOK82qahc4CV0GtzNREM3JsJOpRr4UnoGFW/QMC3I9amIfiMohZoDye3XcWhXLTy+9IYYraN\ne1uR2NsYtkXwbVTziMrd0oC+O1LyeL9GZgV5Z+NOYE4fh9UB/tQ0UaRXjZ5oY3iyezPB65SSxyi3\njV6vjLGTb0dg1sHgKGrDV7WHk9uuDCWF8efaoI/Wy6WBeKgJ20L4NqqJcyOuR82e6iKPt4hzMd6Z\nuKZ7w5zWwflO/RdmM4jSEV6J5tvt6hrzizmhE0bpIVaEnEklyyg8WhvToYtuR9DoSwtK38ZTC+2k\nZCm2K06g6EopLwDberT84q2ywErYFsJ3UBNXhGsVPK27yOPNcS7IOwtXrc7NLizmeAnB4vatF+GZ\nJ5f7glDYRWUBzVNXvJuJPf/vKaP94gORGAWLO3f8TdyOYL4nIDjZteOHi736iz+6NKlRioMffP90\nSCOq5oX3Arpq2s43LYZtIXwH1cQZb/73iRYK8hZxLsY7C3cic7ynhv/0l/BOvjX3ZgLVX0EazRJz\nLWSk5CDlJ+iCyuv3rqcfdqXrZX1l9HPLl7C+M3p/NFq3dooF8QnVcb2CYYvxFnEWLArwTsd1umc4\nZl5aE8Ao7enC+1PKu5mge1NlVO9LH06lEWt0O4IlViw5/kJsiayyo0rX6poyesZqZCxrUoqWgvgC\nNfm0gxYOGqQcXB7vuxMGBXhn4VrdMyQzL3cU+EwnXFOqyrybiSA6AGX0iDGURlwedxAMKXdkEXc5\nZbsgXG8NY0gj+kMnEcLScU5IFcQXqHKgwrXNoMeeN20X03Gt7tmKR7OXFZa5lUZ/Raoq9W4msPys\nNrLuy6URezzfgx0qCfIAXqoquE2ZcK1c2oI0KsXGgrBknEuHPu2o6oL4ApUHqlyFuxrcMeSNLFyr\neybPo0mm5jbM3uPGmU3UfkOqtCGQRuZKVzfI/M9ukcY6xvXeScU3MUYneU1iVjh1hfF916+FN1FL\nwfW7ty9R70DlHGdQqtC3K9OMbBtbDxewY92umKSnGenj8iQfVVfU9WvhTRxScCcy/wrmmyfEmoA1\nhgAAAABJRU5ErkJggg==\n",
       "text": [
        "a_n__(l)\u22c5m_l+1\u22c5\u03b6_n__'(m_l\u22c5x_l) - a_n__(l+1)\u22c5m_l\u22c5\u03b6_n__'(m_l+1\u22c5x_l) - d_n__(l)\u22c5m\n",
        "_l+1\u22c5\u03c8_n__'(m_l\u22c5x_l) + d_n__(l+1)\u22c5m_l\u22c5\u03c8_n__'(m_l+1\u22c5x_l)"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "eqA2.2\n"
       ]
      },
      {
       "latex": [
        "$$b^{(l)}_{n} m_{l+1} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} - b^{(l+1)}_{n} m_{l} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l)}_{n} m_{l+1} \\psi_{n}{\\left (m_{l} x_{l} \\right )} + c^{(l+1)}_{n} m_{l} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAsoAAAAaBAMAAABcGWd1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "text": [
        "b_n__(l)\u22c5m_l+1\u22c5\u03b6_n(m_l\u22c5x_l) - b_n__(l+1)\u22c5m_l\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l)\u22c5m_l+1\u22c5\u03c8\n",
        "_n(m_l\u22c5x_l) + c_n__(l+1)\u22c5m_l\u22c5\u03c8_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "eqA2.3\n"
       ]
      },
      {
       "latex": [
        "$$b^{(l)}_{n} \\zeta^{'}_{n}{\\left (m_{l} x_{l} \\right )} - b^{(l+1)}_{n} \\zeta^{'}_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l)}_{n} \\psi^{'}_{n}{\\left (m_{l} x_{l} \\right )} + c^{(l+1)}_{n} \\psi^{'}_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAlUAAAAaBAMAAABsubQRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZ44TdemzBvVJ4qvOlT4QcvwHGLin1p/d/AwAAAAASUVORK5CYII=\n",
       "text": [
        "b_n__(l)\u22c5\u03b6_n__'(m_l\u22c5x_l) - b_n__(l+1)\u22c5\u03b6_n__'(m_l+1\u22c5x_l) - c_n__(l)\u22c5\u03c8_n__'(m_l\u22c5\n",
        "x_l) + c_n__(l+1)\u22c5\u03c8_n__'(m_l+1\u22c5x_l)"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "eqA2.4\n"
       ]
      },
      {
       "latex": [
        "$$a^{(l)}_{n} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} - a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l)}_{n} \\psi_{n}{\\left (m_{l} x_{l} \\right )} + d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAl0AAAAaBAMAAAB/bvTlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHn0lEQVRYCa2YbYhUVRjH/+PM7J3ZmdmdJKgw\ncaUt+hTzYbVAqZu7aVjhBhX06lSiBWE3XQL7oEOKoBK7En7JiCsYUUGOlC9Q1JTihzIbo3cCh96o\ntGFMZYUQep7zcs+ZO3PXWXcemHuec85z/vd3nnvmzD0DSIvlqayqyhUWVzWPm6GaEesGG9BduvV4\naRKPGsYr8JyCs8/T4x7BDNW0EJVdYAO6TPcWUkXERy3Kabu9HrbyoBJ9HrsJM1RjJWVdYAO6S5dy\nEa/A8TXilZRbgMU8zufLd/REhcOVmVk32IDu0vXm0VcG3pzJzPYAp3m8zxfK18zUWERaN9iA7tL1\nAP15YIQJqVQWsZ9l2n9ti8BZHujzhfPVmRpHKzugnabSZkPSdHWDDjBzidAzN4ThewJ4mDq+5s6C\njoj52guVN4bqopqsITnBns8XzleHahwubZaB102ptys2G+K6A9Oh2/56BF0nsw3uCMO3HVhO7VQA\ng/eoiD5POeGiL9zA9ZSPhM+OuHC+OlTjMcpq2jFlf9Vms/I1LbpLEXRWviL1DAsCvpeBvdROjxLr\njt6nIoZV2VIkgiVodaUG0OtiVr3+Rp22Mc5Xh2qWRps7jnsWGzAniG4TK/sCuheC2NRkezogVtZB\nkXo6gMoghl6XdtGXgdCcP4P+nYEXcpyJUANXkz7SD7Lj84Xz1aEahyvboB1TLoRho9b9Qc/l6b4J\nYuPFZDs66s/kdVCkng6gMuB7EWi4wPsQP5Mygl7IouyfNh1OEf0Jbvf58iN9OlXjeGl9Ve0F5RkY\nNmos6I4O6Ey++ipt6Ugr3oGeDqHkEF/shrvWoMfDkRJws3qFEhHxCrB5/tVL3M3LqYtW3sj8LUPs\n2EGyPnvoIQ9rkL2Nqz59rn9+8DJqHKlsbMm1w67w+Za2Edslw+YsdlFIFGVAmA5GRbwFUpTO1+yh\nldW2dOuPUr6WRujJZr6aifNNt+V7G0iURXcNzqRw+NJLbYV3XOd8KTfB9fU46K5lB/hYFvqa2ol9\ngP6Z8HXzlGo6iHAKmVU9NVHPyCLoI7ZJwxbPNWLVhAIM01kqmk7li+m8dnQonMJcrPDE7cJ6AYM9\nceJzdiPrqy9RrgzrCfdUkayuRewcMg0+Xd6KO7yT7MjdyRLcUMbnlF/VUlHl1GoqiDanuBefzOXF\n2TNXVM2yYLaaYZubLWY8iGcHhOksFd472VS+iO5EW7pY9RgOIS2Dw3qyFUkf1sSJLzuJHhfYAbJ1\nAoM9tnQJjnMRuUnM8vl06eFvaiUHWEkf54/VZGtHyf3KwyrqKZNr2dRq5pzpoM+nYVvpk2zQZSPr\nrv6dPGYrGDYvXaDtpkIdZGE6S4Xp2FS+iO7ptnQOXqX9MCuDw3o0gjqSnxVhTZz40j420Zxz9MEy\nesvnUhopINdAtoi0OF0C56lDLEBNpCJfg3OB3G9VVRWXU/M5TqyFfpc83nVEvqhUxmwlw4bxfByO\nKztb6QIVQXd7vf5svX6GgonuIhWtdEidpXyNRemJ74tc8XrixLepgFOUQW39ee3xike2xk94PO/x\n6TJ5AfQUyZH/QwSBzjnKqxNULWcqNfgcKPK1cpQITlMtVeS2wEJsWEjbTdyT3S10CFR4pbLJ9RVN\nh3gxVqVvm7BWPZOvYOLEt6mK93J5OYauvO3howMjFemmC/yET8SqNPPvEw1agSIFekeVw5xL6K2d\nlH7zdQo1tS1RvnLeCiQ8cfbM1JrGh9hokzyEH5D54oF7W+ksFU2n8hVJRxt3xsuV5HQZ1Z6t2vF4\nfZmJE1+fm3kqeAmht4vdRHz4P9DbW3wAmFfCJ8CvcTpdpi71NuhtmI+ZOM4XY7/h4EDBVI0XrUYx\nPsdRvsar++i1PTlBtWyF2wILsWEb3EwJmfFyOh+ms1U0ncwXIumQmkjgsJpuaLbEoNeXNXHicwYX\nzL5brSgGHfulFBsrgn4F+ctBS3sJHZwX0OnSGVw2/KE8ZsKcAXgI1g3fP1iSj13UzSVSjUJ8DqN8\nrRsZW1SWZ8++ErcFFmbLffnuncA1S9HvhelsFU2n8hVNh59/+iAPOd3QbIlB58uaeMAnV5QATT6H\nWW6Kn/aToi4ve7XPTox7wyYfe7g1Ss0+Z4oxvS4V14WHc91mq3LDIC20CDpWCehUvniIWpTCtS+P\nc0VONzTbW+p/1etlyP2eo3jimk+tKG7mP3bSo9nKKG1jsi6ufLoMnLirKnYhH7vdIvxoNZ/7xX7P\njjh7DrEXMpuNtjmyXTiejKBjlYBuh6UUQUe/4XRrMd2W2er1pVQ4AwGfWlHU5XjbMQ89r1Av/bEZ\nGJ8uhbGzUfnNhXzszW1TqfkcyudMYXz2dGrCDV0MmzztxRp4hl4O2tKxSud0dH7he8nptujJfE1o\nGpq4U9MVtaK4OvIp/bGQWERezOe6tCMly9mj/OZCPvbmtqnUfAoV50wxhM+eNrLRsdgoG0BqAMPl\nCDpWmQYdXBaU022ZLecrduxfEUIuZcDwqRVFzbbNsSvGN/+BmDaSlo/dbmr2w2p+czfV9re0cEN7\nNvuvsKZhV0jXqifWV5O04VMrqql3mhX12DsfVekwtAtswaLs8JYiLNsu+H8DsqSujRwLcQAAAABJ\nRU5ErkJggg==\n",
       "text": [
        "a_n__(l)\u22c5\u03b6_n(m_l\u22c5x_l) - a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l)\u22c5\u03c8_n(m_l\u22c5x_l) + d_\n",
        "n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l)"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"eqA2.1\")\n",
      "display(eqA2_1)\n",
      "#1\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\zeta_n(m_lx_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$\")\n",
      "denom = a__l_n*m_lp1\n",
      "z=m_l*x_l\n",
      "eqA2_1_norm = expand( eqA2_1/(denom * zeta_n(z)) )\n",
      "eqA2_1_norm = eqA2_1_norm.replace( zeta__dash_n(z)/zeta_n(z), D__3_n(z) )\n",
      "eqA2_1_norm = expand( eqA2_1_norm * denom )\n",
      "#display(eqA2_1_norm)\n",
      "#2\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\psi_n(m_lx_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$\")\n",
      "z=m_l*x_l\n",
      "denom = -1 * d__l_n*m_lp1/zeta_n(z)\n",
      "eqA2_1_norm = expand(eqA2_1_norm/(denom*psi_n(z)) )\n",
      "eqA2_1_norm = eqA2_1_norm.replace(psi__dash_n(z)/psi_n(z), D__1_n(z))\n",
      "eqA2_1_norm = expand( eqA2_1_norm * denom )\n",
      "#display(eqA2_1_norm)\n",
      "#3\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\zeta_n(m_{l+1}x_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$\")\n",
      "z=m_l*x_l\n",
      "denom = -1 *a__lp1_n*m_l/(zeta_n(z)*psi_n(z))\n",
      "zp1=m_lp1*x_l\n",
      "eqA2_1_norm = expand( eqA2_1_norm/(denom* zeta_n(zp1)) )\n",
      "eqA2_1_norm = expand(eqA2_1_norm.replace(zeta__dash_n(zp1)/zeta_n (zp1), D__3_n(zp1)))\n",
      "eqA2_1_norm = expand( eqA2_1_norm * denom )\n",
      "#display(eqA2_1_norm)\n",
      "#4\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\psi_n(m_{l+1}x_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$\")\n",
      "z=m_l*x_l\n",
      "zp1=m_lp1*x_l\n",
      "denom =  d__lp1_n*m_l/(zeta_n(z)*zeta_n(zp1)*psi_n(z))\n",
      "eqA2_1_norm = expand(eqA2_1_norm/(denom*psi_n(zp1)) )\n",
      "eqA2_1_norm = eqA2_1_norm.replace(psi__dash_n(zp1)/psi_n(zp1), D__1_n(zp1))\n",
      "eqA2_1_norm = expand( eqA2_1_norm * denom )\n",
      "display(eqA2_1_norm)\n",
      "eqA2_1_result = eqA2_1_norm\n",
      "\n",
      "latex_print(r\"$\\hspace{42pt}$ Check it back\")\n",
      "z=m_l*x_l\n",
      "eqA2_1_norm = eqA2_1_norm.replace( D__1_n(z), psi__dash_n(z)/psi_n(z))\n",
      "eqA2_1_norm = eqA2_1_norm.replace( D__3_n(z), zeta__dash_n(z)/zeta_n(z) )\n",
      "eqA2_1_norm = eqA2_1_norm*zeta_n(z)*psi_n(z)\n",
      "z=m_lp1*x_l\n",
      "eqA2_1_norm = eqA2_1_norm.replace( D__1_n(z), psi__dash_n(z)/psi_n(z))\n",
      "eqA2_1_norm = eqA2_1_norm.replace( D__3_n(z), zeta__dash_n(z)/zeta_n(z) )\n",
      "eqA2_1_norm = eqA2_1_norm*zeta_n(z)*psi_n(z)\n",
      "display(expand(eqA2_1_norm-eqA2_1))\n",
      "\n",
      "        \n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "eqA2.1\n"
       ]
      },
      {
       "latex": [
        "$$a^{(l)}_{n} m_{l+1} \\zeta^{'}_{n}{\\left (m_{l} x_{l} \\right )} - a^{(l+1)}_{n} m_{l} \\zeta^{'}_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l)}_{n} m_{l+1} \\psi^{'}_{n}{\\left (m_{l} x_{l} \\right )} + d^{(l+1)}_{n} m_{l} \\psi^{'}_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAs4AAAAaBAMAAABV8scPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhgAAAABJRU5ErkJggg==\n",
       "text": [
        "a_n__(l)\u22c5m_l+1\u22c5\u03b6_n__'(m_l\u22c5x_l) - a_n__(l+1)\u22c5m_l\u22c5\u03b6_n__'(m_l+1\u22c5x_l) - d_n__(l)\u22c5m\n",
        "_l+1\u22c5\u03c8_n__'(m_l\u22c5x_l) + d_n__(l+1)\u22c5m_l\u22c5\u03c8_n__'(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\zeta_n(m_lx_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1b3e590>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\psi_n(m_lx_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1b10f10>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\zeta_n(m_{l+1}x_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26cc0422d0>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\psi_n(m_{l+1}x_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1b37c10>"
       ]
      },
      {
       "latex": [
        "$$\\frac{a^{(l)}_{n} m_{l+1} \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} - \\frac{a^{(l+1)}_{n} m_{l} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} - \\frac{d^{(l)}_{n} m_{l+1} \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} + \\frac{d^{(l+1)}_{n} m_{l} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "     a_n__(l)\u22c5m_l+1\u22c5D_n__(3)(m_l\u22c5x_l)           a_n__(l+1)\u22c5m_l\u22c5D_n__(3)(m_l+1\u22c5\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",
        "\u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m\n",
        "\n",
        "x_l)           d_n__(l)\u22c5m_l+1\u22c5D_n__(1)(m_l\u22c5x_l)           d_n__(l+1)\u22c5m_l\u22c5D_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\n",
        "_l\u22c5x_l)   \u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_\n",
        "\n",
        "(1)(m_l+1\u22c5x_l)   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "l)\u22c5\u03b6_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Check it back"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1b15890>"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"System to solve:\")\n",
      "print(\"Eq1:\")\n",
      "display(eqA2_1_result)\n",
      "print(\"Eq2:\")\n",
      "display(eqA2_4)\n",
      "result_ad = solve([eqA2_1_result, eqA2_4],[a__l_n, d__l_n])\n",
      "result_a__l_n = result_ad.items()[0][1]\n",
      "display(Latex(r\"Result for  $a^{(l)}_n$:\"))\n",
      "display(simplify(result_a__l_n))\n",
      "result_d__l_n = result_ad.items()[1][1]\n",
      "display(Latex(r\"Result for  $d^{(l)}_n$:\"))\n",
      "display(simplify(result_d__l_n))\n",
      "print(\"Check the solution\")\n",
      "eq1=eqA2_1_result.subs(a__l_n, result_a__l_n).subs(d__l_n, result_d__l_n)\n",
      "display(simplify(eq1))\n",
      "eq1=eqA2_4.subs(a__l_n, result_a__l_n).subs(d__l_n, result_d__l_n)\n",
      "display(simplify(eq1))\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "System to solve:\n",
        "Eq1:\n"
       ]
      },
      {
       "latex": [
        "$$\\frac{a^{(l)}_{n} m_{l+1} \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} - \\frac{a^{(l+1)}_{n} m_{l} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} - \\frac{d^{(l)}_{n} m_{l+1} \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} + \\frac{d^{(l+1)}_{n} m_{l} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "     a_n__(l)\u22c5m_l+1\u22c5D_n__(3)(m_l\u22c5x_l)           a_n__(l+1)\u22c5m_l\u22c5D_n__(3)(m_l+1\u22c5\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",
        "\u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m\n",
        "\n",
        "x_l)           d_n__(l)\u22c5m_l+1\u22c5D_n__(1)(m_l\u22c5x_l)           d_n__(l+1)\u22c5m_l\u22c5D_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\n",
        "_l\u22c5x_l)   \u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_\n",
        "\n",
        "(1)(m_l+1\u22c5x_l)   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "l)\u22c5\u03b6_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Eq2:\n"
       ]
      },
      {
       "latex": [
        "$$a^{(l)}_{n} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} - a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l)}_{n} \\psi_{n}{\\left (m_{l} x_{l} \\right )} + d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAl0AAAAaBAMAAAB/bvTlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "text": [
        "a_n__(l)\u22c5\u03b6_n(m_l\u22c5x_l) - a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l)\u22c5\u03c8_n(m_l\u22c5x_l) + d_\n",
        "n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "Result for  $a^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1bf7950>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5D_n__(1)(m_l+\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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5\u03c8_n(m_l\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03b6_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "+1\u22c5x_l))\u22c5D_n__(1)(m_l\u22c5x_l)\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\n",
        "                          "
       ]
      },
      {
       "latex": [
        "Result for  $d^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1bb3f50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\psi_{n}{\\left (m_{l} x_{l} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5D_n__(1)(m_l+\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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5\u03c8_n(m_l\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03c8_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "+1\u22c5x_l))\u22c5D_n__(3)(m_l\u22c5x_l)\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\n",
        "                          "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Check the solution\n"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"eqA2.3\")\n",
      "display(eqA2_3)\n",
      "#1\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\zeta_n(m_lx_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$\")\n",
      "denom = b__l_n\n",
      "z=m_l*x_l\n",
      "eqA2_3_norm = expand( eqA2_3/(denom * zeta_n(z)) )\n",
      "eqA2_3_norm = eqA2_3_norm.replace( zeta__dash_n(z)/zeta_n(z), D__3_n(z) )\n",
      "eqA2_3_norm = expand( eqA2_3_norm * denom )\n",
      "#display(eqA2_3_norm)\n",
      "#2\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\psi_n(m_lx_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$\")\n",
      "z=m_l*x_l\n",
      "denom = -1 * c__l_n/zeta_n(z)\n",
      "eqA2_3_norm = expand(eqA2_3_norm/(denom*psi_n(z)) )\n",
      "eqA2_3_norm = eqA2_3_norm.replace(psi__dash_n(z)/psi_n(z), D__1_n(z))\n",
      "eqA2_3_norm = expand( eqA2_3_norm * denom )\n",
      "#display(eqA2_3_norm)\n",
      "#3\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\zeta_n(m_{l+1}x_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$\")\n",
      "z=m_l*x_l\n",
      "denom = -1 *b__lp1_n/(zeta_n(z)*psi_n(z))\n",
      "zp1=m_lp1*x_l\n",
      "eqA2_3_norm = expand( eqA2_3_norm/(denom* zeta_n(zp1)) )\n",
      "eqA2_3_norm = expand(eqA2_3_norm.replace(zeta__dash_n(zp1)/zeta_n (zp1), D__3_n(zp1)))\n",
      "eqA2_3_norm = expand( eqA2_3_norm * denom )\n",
      "#display(eqA2_3_norm)\n",
      "#4\n",
      "latex_print(r\"$\\hspace{42pt}$ Divide with $\\psi_n(m_{l+1}x_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$\")\n",
      "z=m_l*x_l\n",
      "zp1=m_lp1*x_l\n",
      "denom =  c__lp1_n/(zeta_n(z)*zeta_n(zp1)*psi_n(z))\n",
      "eqA2_3_norm = expand(eqA2_3_norm/(denom*psi_n(zp1)) )\n",
      "#display(eqA2_3_norm)\n",
      "eqA2_3_norm = eqA2_3_norm.replace(psi__dash_n(zp1)/psi_n(zp1), D__1_n(zp1))\n",
      "eqA2_3_norm = expand( eqA2_3_norm * denom )\n",
      "display(eqA2_3_norm)\n",
      "eqA2_3_result = eqA2_3_norm\n",
      "\n",
      "latex_print(r\"$\\hspace{42pt}$ Check it back\")\n",
      "z=m_l*x_l\n",
      "eqA2_3_norm = eqA2_3_norm.replace( D__1_n(z), psi__dash_n(z)/psi_n(z))\n",
      "eqA2_3_norm = eqA2_3_norm.replace( D__3_n(z), zeta__dash_n(z)/zeta_n(z) )\n",
      "eqA2_3_norm = eqA2_3_norm*zeta_n(z)*psi_n(z)\n",
      "z=m_lp1*x_l\n",
      "eqA2_3_norm = eqA2_3_norm.replace( D__1_n(z), psi__dash_n(z)/psi_n(z))\n",
      "eqA2_3_norm = eqA2_3_norm.replace( D__3_n(z), zeta__dash_n(z)/zeta_n(z) )\n",
      "eqA2_3_norm = eqA2_3_norm*zeta_n(z)*psi_n(z)\n",
      "display(expand(eqA2_3_norm-eqA2_3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "eqA2.3\n"
       ]
      },
      {
       "latex": [
        "$$b^{(l)}_{n} \\zeta^{'}_{n}{\\left (m_{l} x_{l} \\right )} - b^{(l+1)}_{n} \\zeta^{'}_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l)}_{n} \\psi^{'}_{n}{\\left (m_{l} x_{l} \\right )} + c^{(l+1)}_{n} \\psi^{'}_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAlUAAAAaBAMAAABsubQRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZ44TdemzBvVJ4qvOlT4QcvwHGLin1p/d/AwAAAAASUVORK5CYII=\n",
       "text": [
        "b_n__(l)\u22c5\u03b6_n__'(m_l\u22c5x_l) - b_n__(l+1)\u22c5\u03b6_n__'(m_l+1\u22c5x_l) - c_n__(l)\u22c5\u03c8_n__'(m_l\u22c5\n",
        "x_l) + c_n__(l+1)\u22c5\u03c8_n__'(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\zeta_n(m_lx_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1a16650>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\psi_n(m_lx_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1ab5a50>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\zeta_n(m_{l+1}x_l)$ and substitute with $D^{(3)}_n=\\zeta^{'}_n/\\zeta_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1ab5a50>"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Divide with $\\psi_n(m_{l+1}x_l)$ and substitute with $D^{(1)}_n=\\psi^{'}_n/\\psi_n$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1abda90>"
       ]
      },
      {
       "latex": [
        "$$\\frac{b^{(l)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} - \\frac{b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} - \\frac{c^{(l)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} + \\frac{c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "        b_n__(l)\u22c5D_n__(3)(m_l\u22c5x_l)                b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_\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",
        "\u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m\n",
        "\n",
        "l)                c_n__(l)\u22c5D_n__(1)(m_l\u22c5x_l)                c_n__(l+1)\u22c5D_n__(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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "_l\u22c5x_l)   \u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_\n",
        "\n",
        ")(m_l+1\u22c5x_l)     \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "l)\u22c5\u03b6_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "$\\hspace{42pt}$ Check it back"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1abdbd0>"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(\"System to solve:\")\n",
      "print(\"Eq1:\")\n",
      "display(eqA2_3_result)\n",
      "print(\"Eq2:\")\n",
      "display(eqA2_2)\n",
      "result_bc = solve([eqA2_3_result, eqA2_2],[b__l_n, c__l_n])\n",
      "result_b__l_n = result_bc.items()[0][1]\n",
      "display(Latex(r\"Result for  $b^{(l)}_n$:\"))\n",
      "display(simplify(result_b__l_n))\n",
      "result_c__l_n = result_bc.items()[1][1]\n",
      "display(Latex(r\"Result for  $c^{(l)}_n$:\"))\n",
      "display(simplify(result_c__l_n))\n",
      "print(\"Check the solution\")\n",
      "eq1=eqA2_3_result.subs(b__l_n, result_b__l_n).subs(c__l_n, result_c__l_n)\n",
      "display(simplify(eq1))\n",
      "eq1=eqA2_2.subs(b__l_n, result_b__l_n).subs(c__l_n, result_c__l_n)\n",
      "display(simplify(eq1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "System to solve:\n",
        "Eq1:\n"
       ]
      },
      {
       "latex": [
        "$$\\frac{b^{(l)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} - \\frac{b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} - \\frac{c^{(l)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}} + \\frac{c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )}}{\\psi_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "        b_n__(l)\u22c5D_n__(3)(m_l\u22c5x_l)                b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_\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",
        "\u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m\n",
        "\n",
        "l)                c_n__(l)\u22c5D_n__(1)(m_l\u22c5x_l)                c_n__(l+1)\u22c5D_n__(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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "_l\u22c5x_l)   \u03c8_n(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l)   \u03c8_n(m_l\u22c5x_l)\u22c5\u03b6_n(m_l\u22c5x_\n",
        "\n",
        ")(m_l+1\u22c5x_l)     \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "l)\u22c5\u03b6_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Eq2:\n"
       ]
      },
      {
       "latex": [
        "$$b^{(l)}_{n} m_{l+1} \\zeta_{n}{\\left (m_{l} x_{l} \\right )} - b^{(l+1)}_{n} m_{l} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l)}_{n} m_{l+1} \\psi_{n}{\\left (m_{l} x_{l} \\right )} + c^{(l+1)}_{n} m_{l} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAsoAAAAaBAMAAABcGWd1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "text": [
        "b_n__(l)\u22c5m_l+1\u22c5\u03b6_n(m_l\u22c5x_l) - b_n__(l+1)\u22c5m_l\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l)\u22c5m_l+1\u22c5\u03c8\n",
        "_n(m_l\u22c5x_l) + c_n__(l+1)\u22c5m_l\u22c5\u03c8_n(m_l+1\u22c5x_l)"
       ]
      },
      {
       "latex": [
        "Result for  $b^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1ab5a50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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HqxyqchxUaT66ZPxLUH0ai9wn+M2kBCJn0pM5H9SjZeoUlb5HMAeLgM2t5FK/LYc5V0Ss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\n",
       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l))\u22c5D_n__(1)(m_l\u22c5x_l) \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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "- m_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5D_n__(1)(m\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03b6_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "_l+1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l))\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",
        "                         "
       ]
      },
      {
       "latex": [
        "Result for  $c^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b19d2ed0>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\psi_{n}{\\left (m_{l} x_{l} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l))\u22c5D_n__(3)(m_l\u22c5x_l) \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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "- m_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5D_n__(1)(m\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03c8_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "_l+1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l))\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",
        "                         "
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Check the solution\n"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      },
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally\n",
      "------"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print (\"l = [1..L] or in C++ l = [0..L-1]\")\n",
      "display(Latex(r\"Result for  $a^{(l)}_n$:\"))\n",
      "display(simplify(result_a__l_n))\n",
      "display(Latex(r\"Result for  $d^{(l)}_n$:\"))\n",
      "display(simplify(result_d__l_n))\n",
      "display(Latex(r\"Result for  $b^{(l)}_n$:\"))\n",
      "display(simplify(result_b__l_n))\n",
      "display(Latex(r\"Result for  $c^{(l)}_n$:\"))\n",
      "display(simplify(result_c__l_n))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "l = [1..L] or in C++ l = [0..L-1]\n"
       ]
      },
      {
       "latex": [
        "Result for  $a^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1be1710>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5D_n__(1)(m_l+\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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5\u03c8_n(m_l\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03b6_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "+1\u22c5x_l))\u22c5D_n__(1)(m_l\u22c5x_l)\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\n",
        "                          "
       ]
      },
      {
       "latex": [
        "Result for  $d^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1a64650>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\psi_{n}{\\left (m_{l} x_{l} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5D_n__(1)(m_l+\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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l)) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - d_n__(l+1)\u22c5\u03c8_n(m_l\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03c8_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "+1\u22c5x_l))\u22c5D_n__(3)(m_l\u22c5x_l)\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\n",
        "                          "
       ]
      },
      {
       "latex": [
        "Result for  $b^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1be1750>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\zeta_{n}{\\left (m_{l} x_{l} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l))\u22c5D_n__(1)(m_l\u22c5x_l) \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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "- m_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5D_n__(1)(m\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03b6_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "_l+1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l))\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",
        "                         "
       ]
      },
      {
       "latex": [
        "Result for  $c^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1bd1c50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (m_{l} x_{l} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )}\\right) \\psi_{n}{\\left (m_{l} x_{l} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (m_{l} x_{l} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\zeta_{n}{\\left (m_{l+1} x_{l} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (m_{l+1} x_{l} \\right )} \\psi_{n}{\\left (m_{l+1} x_{l} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5\u03c8_n(m_l+1\u22c5x_l))\u22c5D_n__(3)(m_l\u22c5x_l) \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",
        "                                                              m_l+1\u22c5(D_n__(1)(\n",
        "\n",
        "- m_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(m_l+1\u22c5x_l)\u22c5\u03b6_n(m_l+1\u22c5x_l) - c_n__(l+1)\u22c5D_n__(1)(m\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",
        "m_l\u22c5x_l) - D_n__(3)(m_l\u22c5x_l))\u22c5\u03c8_n(m_l\u22c5x_l)                                    \n",
        "\n",
        "_l+1\u22c5x_l)\u22c5\u03c8_n(m_l+1\u22c5x_l))\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",
        "                         "
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print (\"in C++ l = [0..L-1] zl=ml*xl zlp1=m(l+1)xl\")\n",
      "z, z1 = symbols('z__(l), z1__(l)');\n",
      "display(Latex(r\"Result for  $a^{(l)}_n$:\"))\n",
      "c_a__l_n=simplify(result_a__l_n).replace(m_l*x_l, z).replace(m_lp1*x_l, z1)\n",
      "display(c_a__l_n)\n",
      "display(Latex(r\"Result for  $d^{(l)}_n$:\"))\n",
      "c_d__l_n=simplify(result_d__l_n).replace(m_l*x_l, z).replace(m_lp1*x_l, z1)\n",
      "display(c_d__l_n)\n",
      "display(Latex(r\"Result for  $b^{(l)}_n$:\"))\n",
      "c_b__l_n=simplify(result_b__l_n).replace(m_l*x_l, z).replace(m_lp1*x_l, z1)\n",
      "display(c_b__l_n)\n",
      "display(Latex(r\"Result for  $c^{(l)}_n$:\"))\n",
      "c_c__l_n=simplify(result_c__l_n).replace(m_l*x_l, z).replace(m_lp1*x_l, z1)\n",
      "display(c_c__l_n)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "in C++ l = [0..L-1] zl=ml*xl zlp1=m(l+1)xl\n"
       ]
      },
      {
       "latex": [
        "Result for  $a^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26cc042a50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )}\\right) \\zeta_{n}{\\left (z^{(l)} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(z_1__(l))\u22c5\u03b6_n(z_1__(l)) - d_n__(l+1)\u22c5D_n__(1)(z_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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                            m_l+1\u22c5(D_n__(1)(z_\n",
        "\n",
        "l))\u22c5\u03c8_n(z_1__(l))) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(z_1__(l)) - d_n__(l+1)\u22c5\u03c8_n(z_1__(l)\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",
        "_(l)) - D_n__(3)(z__(l)))\u22c5\u03b6_n(z__(l))                                         \n",
        "\n",
        "))\u22c5D_n__(1)(z__(l))\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                   "
       ]
      },
      {
       "latex": [
        "Result for  $d^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1be9a50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )}\\right) \\psi_{n}{\\left (z^{(l)} \\right )}} \\left(- m_{l} \\left(a^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - d^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) + m_{l+1} \\left(a^{(l+1)}_{n} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - d^{(l+1)}_{n} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "-m_l\u22c5(a_n__(l+1)\u22c5D_n__(3)(z_1__(l))\u22c5\u03b6_n(z_1__(l)) - d_n__(l+1)\u22c5D_n__(1)(z_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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                                                            m_l+1\u22c5(D_n__(1)(z_\n",
        "\n",
        "l))\u22c5\u03c8_n(z_1__(l))) + m_l+1\u22c5(a_n__(l+1)\u22c5\u03b6_n(z_1__(l)) - d_n__(l+1)\u22c5\u03c8_n(z_1__(l)\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",
        "_(l)) - D_n__(3)(z__(l)))\u22c5\u03c8_n(z__(l))                                         \n",
        "\n",
        "))\u22c5D_n__(3)(z__(l))\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                   "
       ]
      },
      {
       "latex": [
        "Result for  $b^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1a16d50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )}\\right) \\zeta_{n}{\\left (z^{(l)} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(z_1__(l)) - c_n__(l+1)\u22c5\u03c8_n(z_1__(l)))\u22c5D_n__(1)(z__(l)) - m\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",
        "                                                           m_l+1\u22c5(D_n__(1)(z__\n",
        "\n",
        "_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(z_1__(l))\u22c5\u03b6_n(z_1__(l)) - c_n__(l+1)\u22c5D_n__(1)(z_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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(l)) - D_n__(3)(z__(l)))\u22c5\u03b6_n(z__(l))                                          \n",
        "\n",
        "l))\u22c5\u03c8_n(z_1__(l)))\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                  "
       ]
      },
      {
       "latex": [
        "Result for  $c^{(l)}_n$:"
       ],
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Latex at 0x7f26b1bb3f50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{1}{m_{l+1} \\left(\\operatorname{D^{(1)}_{n}}{\\left (z^{(l)} \\right )} - \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )}\\right) \\psi_{n}{\\left (z^{(l)} \\right )}} \\left(m_{l} \\left(b^{(l+1)}_{n} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - c^{(l+1)}_{n} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right) \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)} \\right )} - m_{l+1} \\left(b^{(l+1)}_{n} \\operatorname{D^{(3)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\zeta_{n}{\\left (z^{(l)}_{1} \\right )} - c^{(l+1)}_{n} \\operatorname{D^{(1)}_{n}}{\\left (z^{(l)}_{1} \\right )} \\psi_{n}{\\left (z^{(l)}_{1} \\right )}\\right)\\right)$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "m_l\u22c5(b_n__(l+1)\u22c5\u03b6_n(z_1__(l)) - c_n__(l+1)\u22c5\u03c8_n(z_1__(l)))\u22c5D_n__(3)(z__(l)) - m\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",
        "                                                           m_l+1\u22c5(D_n__(1)(z__\n",
        "\n",
        "_l+1\u22c5(b_n__(l+1)\u22c5D_n__(3)(z_1__(l))\u22c5\u03b6_n(z_1__(l)) - c_n__(l+1)\u22c5D_n__(1)(z_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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(l)) - D_n__(3)(z__(l)))\u22c5\u03c8_n(z__(l))                                          \n",
        "\n",
        "l))\u22c5\u03c8_n(z_1__(l)))\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                  "
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}