{ "metadata": { "name": "", "signature": "sha256:18bc049af5dffe6f15c77ca07acc3f7e88a474c9495ab48a8f77594110bf41f9" }, "nbformat": 3, "nbformat_minor": 0, "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": 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"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": 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"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": 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"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": 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"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": 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"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": [ "" ] }, { "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": [ "" ] }, { "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": [ "" ] }, { "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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABJEAAAA5BAMAAAB0YKcRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSBlUW0Hz/xvwgAOeoJvBAAAAAElFTkSuQmCC\n", "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": [ "" ] }, { "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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rJWwwNG5ujsxyPbViGw3GBoOPzxk37nwwuSB1huOOE+DSVn95AJuyf9HJwsDSQjwSlgQY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\nSBlUW0Hz/xvwgAOeoJvBAAAAAElFTkSuQmCC\n", "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": [ "" ] }, { "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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABkMAAABABAMAAABbxR2tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n", "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": [ "" ] }, { "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": [ "" ] }, { "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": [ "" ] }, { "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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABJEAAAA5BAMAAAB0YKcRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nR24kKl7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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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABJEAAAA5BAMAAAB0YKcRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nF0+U7VlTYc82jw7aYVImNOYueZz/A7YSTQ8bsKyqAAAAAElFTkSuQmCC\n", "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": [ "" ] }, { "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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABiwAAABABAMAAAB7KUJQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwkXMIgqGq6sFrfCA9sD/A2AU6TVJuYj7AAAAAElFTkSuQmCC\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__(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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABj8AAABABAMAAAC3sHi7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ng/ZbsYa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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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABkMAAABABAMAAABbxR2tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n", "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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABicAAABABAMAAACDybmnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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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": [ "" ] }, { "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": "iVBORw0KGgoAAAANSUhEUgAABiwAAABABAMAAAB7KUJQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwkXMIgqGq6sFrfCA9sD/A2AU6TVJuYj7AAAAAElFTkSuQmCC\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__(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": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "print (\"in C++ l = [0..L-1] zl=ml*xl zlp1=m(l+1)xl\")\n", "z__l, z__l_1, m__l_1, m__l = symbols('z__(l), z__(l)_1, m__(l)_1, m__(l)');\n", "D__l_1zn, D__l_1z1n, D__l_3zn, D__l_3z1n = symbols('D__(l)_1zn, D__(l)_1z1n, D__(l)_3zn, D__(l)_3z1n');\n", "psi__l_zn, psi__l_z1n, zeta__l_zn, zeta__l_z1n = symbols('psi__(l)_zn, psi__(l)_z1n, zeta__(l)_zn, zeta__(l)_z1n');\n", "c_a__l_n=simplify(result_a__l_n);\n", "c_b__l_n=simplify(result_b__l_n);\n", "c_c__l_n=simplify(result_c__l_n);\n", "c_d__l_n=simplify(result_d__l_n);\n", "all_eq = [c_a__l_n, c_d__l_n, c_b__l_n, c_c__l_n]\n", "for i, eq in enumerate(all_eq):\n", " all_eq[i]= all_eq[i].replace(m_l*x_l, z__l).replace(m_lp1*x_l, z__l_1)\n", " all_eq[i]= all_eq[i].replace(D__3_n(z__l),D__l_3zn).replace(D__1_n(z__l),D__l_1zn)\n", " all_eq[i]= all_eq[i].replace(D__3_n(z__l_1),D__l_3z1n).replace(D__1_n(z__l_1),D__l_1z1n)\n", " all_eq[i]= all_eq[i].replace(zeta_n(z__l),zeta__l_zn).replace(psi_n(z__l),psi__l_zn)\n", " all_eq[i]= all_eq[i].replace(zeta_n(z__l_1),zeta__l_z1n).replace(psi_n(z__l_1),psi__l_z1n)\n", " all_eq[i]= all_eq[i].replace(m_lp1, m__l_1).replace(m_l, m__l)\n", "display(Latex(r\"Result for $a^{(l)}_n$:\"))\n", "display(all_eq[0])\n", "display(Latex(r\"Result for $d^{(l)}_n$:\"))\n", "display(all_eq[1])\n", "display(Latex(r\"Result for $b^{(l)}_n$:\"))\n", "display(all_eq[2])\n", "display(Latex(r\"Result for $c^{(l)}_n$:\"))\n", "display(all_eq[3])" ], "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": [ "" ] }, { "latex": [ "$$\\frac{1}{m^{(l)}_{1} \\zeta^{(l)}_{zn} \\left(D^{(l)}_{1zn} - D^{(l)}_{3zn}\\right)} \\left(D^{(l)}_{1zn} m^{(l)}_{1} \\left(a^{(l+1)}_{n} \\zeta^{(l)}_{z1n} - d^{(l+1)}_{n} \\psi^{(l)}_{z1n}\\right) - m^{(l)} \\left(- D^{(l)}_{1z1n} d^{(l+1)}_{n} \\psi^{(l)}_{z1n} + D^{(l)}_{3z1n} a^{(l+1)}_{n} \\zeta^{(l)}_{z1n}\\right)\\right)$$" ], "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAA9IAAABABAMAAAAE+1I4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZkSJ\nVDL+6OvUAAAACXBIWXMAAA7EAAAOxAGVKw4bAAARLElEQVR4Ae1cfYxcVRU/+2Z2Z3dnZ2aXBCto\nZFBSStTsRBRBgjsIFlMTWKKiENBBoPWDhKkCjaXa5UNJGtIsIBr8fPiRVG3smBC+gunwR6loJJOY\nEKgJHRQjGHWXgoCFgufcz3Pve2/2zc52apO5Se8999zzO+fce+698+bN6QIspxyzHNAAc/StwPsP\nHH0+DzxexgqcdNEg0stYtqMRkhtE+mgM2zJ87i3Sv+zW4njYLWKZ8qu7wHU9iy50r5xoUCNdsl6O\n1p4inZtkJk9FusD6ksxUHPbGiIDPEICWz03V5x7kuWud0c4stCi5EZ2NHo1tuflYAYAUIi6SA34j\nhmTtSqXq9RTpB8rWRtAMLigXQ8uQ1HvgpANg2Pmb/PFIXwD+EWGnYLgeXJ8CIUX4LAyI3DBuG24n\nwjUfK5lCxMU5gFyNBmXtiqXq9RLpYDszkS3DgwCPM44gPwL5ecte09bjtFlZsYdYACbabMwlXSA/\ndq4HQ6GL4z3n3nBmYaSEG5HZgEAaIefUu+atDKOSRBK1uoDjhSpZM60pyV4iPTLLjNwPsAfgJMYh\nMl+HCZTS7Kv0sNisuvMsgDnEEhBU9ZjfcmCxam8LlHM9yHR41HTuDWcW2px0Q7utuQACqbodzVsI\no1wP7UCiVhcwUyaIrC04LdVLpBfKzMo5ALjbsm3GQjJbgaHQsItm9eVmBWigzN++CGAOsQKc76qx\nPQXMI6d40by9LbDvebDegnzKuTecWWhJ6YY/GwB5RUEK81oTbz0PzVCiVhcwUiOErA02NdFLpI/l\nVnDVX8RPkTrnAYwBlCqGPTSrR+VmBagS4yl8pKxiS0UBNsletFbALI0U0CY7dtjjHsRGUCh07w1n\nFtqgdMOfjb6iIIV5rYm3nod6SF18MVpdgDwo9rhofLq2h0hncF1NKU5CcQ4Xv2Y4gngG4DkkFHum\nokflZrWRhvPViAI8oQX9VgHNUmfbWsL3YKiuR/zWuTecWRhJ6YY/G31F2ZgkmzeqLOF7qEeUP1Gt\nPmCVQMhag1O3PUR6GHecKfkq5Kr4uYz/eLkX4FrsK/bFZkhuVhZpfYgVAJv4ooAm0vbY+R6Mz8Vr\n8O4NZxYGId3wZ6NvHBuTZPNGlSV8D/XImLz4olp9wA6BkLUGp22Hv/DfL6WV9eVG+aHJ1yCLfdyF\nTrkP4DxkKPatekxtVhZpcYgrAAqAZyq2aKCJtDp2RdxNngeZ12I1INO5N5xZGIR0w5+NRtqYJJs3\nqizhe6hHlD+e1kx0SjMNgshag/vRzrTJyvVvrr1ybQXwQXT038I3x/SzANtRDOeIJfMG1VTUZmWR\nFoe4CaAAuNIx5d5vaqCJNDKoTOBu8j24TYzEVM69IWehpXLXvLT+3Cu0G9JtPZb/2KxC2ph0MK9h\n2CqtEQ+VSLzWmCmNNgkha4XtS7NLWnkBYPwQBPNQyiHVdE2fArCIR12yxxf1oNzdI1NTN07hAzs+\nkdFpAdiwGRRgp5Z020MSePnUW6emQvFEpo4dLkvEg8tdrO0594aahR4tTQLsaik3vNmUWhKZzrzW\nSK3UGvFQicRrHdkU+oARdA4fvkWtsH1p5Kdu8XU09hmAO2D4D+hFwzU9VoZLkSXZ9iOxKM8fO9N0\niFd/+uugAF921ahe/oAG2jNdE2Mfwtr3YKtCRRrn3rDPDkJuoQkwPKfc8Gazu6yQ7EzXBCrOPO69\nihjESmn1PVTDsVqDqza1/CnlFgkha4XtS/NtYSVHH4e7Q9gsekOe5VzI2GNVPao2q4r0k8jeiQvz\nNmwVYFJLOu3EvAbKSM/p2wJ+inK+B/vNQjtKwL035CyMxNYGfrYcVG54s3lMI1WkO5vHbxx1rVZp\n9T1Uw+oec7WeiI74UyqIbzuy1qr70f5HGMkuYjPTlG7CPt9wVTAkuzRrRuXulpH+wI83AOAhHgpp\nuEpVIaQ6UoZm1S4X1jIXv1zXl0gTZUX0mQcLrYgCyXDuDTkLI3kBUhm8pqrE8WZzrL5xUpnnkVZa\nfQ/JBhblj6v1eBrxAMVDxJQ1Uf0qrwhDpSo2u2uQD7ENalhhefiKD15ff/ha3JYPUVexS3XqnLb3\n+bLarGpBiTspXqYiIQCxvzmethdjt5mE9RIggccu2FOHZm7e9QCHpkOsWDFOOfeGnIURewmp/MvS\nDT0bMZi5+huHQCFVBJDfwTyPtNLqrJHWekes1olZGvcAwUFiypqoPhVlcIHCt7UJ8DS2uTJWWJrn\n1oNXGoU5nG0bu4otlj5/M+AOl5sVQMwHJfAQB/JVqQB8BVl+IWBZA4f16D587i4sZlr0XoB7gONy\nY2lB7lSVmGgSi7dsRQp8Ds80uaFnQ3LwQCWLJqqChjTmUQctDRWj1fNQaF2M1TpdFkgPILeltzmF\n4OGsMmKDwdY2Grmm4Vgqtm6nK9A+a8vRmRDbNSF8Um1WhsFDLHcx43kkAi+MAOnYnTU8P16GOU8c\nI910WMyph2gATWJRsxA0Vrk3sBqJ+SYebIPhSXXjaGHcJ7Vk8yzSHbVWY7XusUYYRU+/AK/Dm30r\nZE99XNBHELxKlS1B8BoUDsBI1bKIEh+cnyvDd5GmzcoKHuKxFutrMjjubiy3t7GPwB9g4wLp2JVH\nmxP2dkARVeRjwYmk4O53YFSsU+zekLMwQuKpozSnVdh2+ACMNeVRt0xx6pPMs0h31FqP1fpDZsWS\n8qHCe7Sww4eJKsozTR9B40g+y80UFgG/UtHrJ85eCFHmFgje4KKGLrU1iV+lYgoCY84aCe6uTECA\ntrBwKHsAFGPWKdGVlZqF5oinjl2h1aP9H63CdENLuW28+YmpqeNunJqaJVmtde+jZQnlWtuS5dV2\njbQsCRyZSMt7T3wEramKnx6ts3jR4QnYXSnTL5Km0O0dvA6FxcCwGFGqqI74NZINSDIZCI/BWTBR\nJjEH6t3edPtKp6RCWXu3Nz11BLdYPcb/6SbshzJHGjrBPDvTSutwOFwVqBRaza9HRpaQ+vYWWvpW\nBeLBQHwEnY1Gn+KGR5t0Ai7MtBz2NC3jIchO/onLajobaqowrynWJgPxc/8n8GcpyqGlkMGRtE4x\nvpyFYdBTR3YSg6Rd0NOabsHZhYqR40SCeRZppXW4qZ9clta6zljQssSQ16isjUAHYjzsMNh5iB5B\ndBEbbGgR4MFZZHGH6CX8LoC348cnZ4tDtgp+UWuOX/L367Qa3Wa2acoss2aINhEID0B9vCFlOXSm\n5eCZU5wvj4nmnIDPfHdhx+jR/g/Vx78zAT//9Saaq1sSzLNIa63ioY7AS2s9PdRGtCz2M+L7tKz1\n8KlIFHTHz0ncaAdkYpS3JGxYkFwZS038Fg6OnPDy+is/S1LoUOb2dzVkDB/EXz4AnnjczolERmtY\nrd7y1w2N8d3haIVYvJz8loZQQcscsxWSgYXL1n0VFZ1xzJkOdKHBteOGNE5xPs3ClMvfXL/+o23s\nUqRJH01LhDfY8PhpN8CvDsLNRlgTCeZtpI1WWNNOqzXYiO+SHA/QXFE8qMhaWRfZVsVQ++J2nQRN\nJ6VKy7utg87VzCDd2bbgkgzv2wt+DJFtCl2Lsrzva1Aq645pi3cJFSJccVsBBROALdJx2c66A91f\nJu5SxZ2FlqZIkz6KtA1v5uR5wC/VfinGmreRNvInr+pC6/OzEQ/w+wwWWSulWTdR0+3ixrLFSamy\nbE656OPN0O8MRQSFNGhFQsEjPTFHgqJswCs3Wp4TKihc8RFFRCyQvmthea84iwbq+ickYqp4KYo0\n6cNp8fCO1PNzUR3x5iPf/RFIT2RptRbRkicrF1DWyo373URNt3uVEqLGTaliA4x00TNlPeSeGQop\n3nZ+KHik2WbcDp8qaj26Dcr3ShW0zLERRck4IODzAJZMQ0TaQNcK7lKVOwstTS6QPtrALLyj7eHZ\ntpYxbbx5M8wI/PaeVus789WIbLZGumSttJ7jJmo6XSfjLCtzN89XuNjGQbPUxOmQi+OSZBrFsh8K\nHmn75jGzCN9/mqMFvekTUgUtc2xE0UQsMCfgZ8JfxOerhqZ7RHVnoX0iF0gfRZqFdwbGftvWMqaN\nN2+GDbGmjd/p0mrNXVeJyJZC0iVrpRX9fBFf7dXjukOzikvNGAIrAJsYK0K6yuxGcfYWPIm/Xfzx\nZ5FQINsWc8ryNdgSWr6lSAUU5iJqtEQiEDfB1fuqHFrEFUhR3FloALlA+mhaLLz3Qe73WsRrffPe\nMHXHKuM3daM1IruAocJXjaImCp/R3ERNt2sTNFHyGZm7+YSAxVcuGmCVFhuf0xS29NNj/bljwAuF\n+EXSiO0wVBJBKujXSE9Nkjjn54o3ljl0YpKPJtLOLLQU6SF9YlodwqvlsfXNsyFNFvc+WulGa0T2\nEaFJ1lIpXvG5Kn4I4z8qbvdiyZQ1fjJeixQ2icVFA5hw4buMLst0q0tAL+KjYSp097NIpfawCG0T\nWmUtDeCJsImaFTpntgu3cifuk7mbeLQTi4vmqYn8e3kinA/k0h0zDlk+7XxudFDT9Sw66Dq8Q4U5\n0i9rZQnf89tEzSa9rrddlaCpUjrx9fn2tsi9VVDWJGR9jja1zGhbU2nb7WkFV0Dueyl1dD+LlIpX\nXGwsJJWyVsoDnqi5YbPT1a9eX8CfoA7pxKidCug2UsRBo8CIOZjjhnJxyb17kodWeiS1c6kFV9rD\nrvXJ1XPXkCUhUsIl69KviVRUSueYzN3EN2/RkpD1yVITz4uCOnPoQ6RP5fRKWkNdzyKt4hWWy4gX\nsbI2qjcLaghrkXBpu3j4q2JMpXTmQtGLPZwJWZ8sNfGUtkCnr4IfpZftTTKI/yk/TmnXs4hT0gfe\nKS0yImtjLiuofVgPhVjZLn59nhVjIg1ipglV6hVCqv2iRTgaZVhqYuZOH7NU/4zUJ20pTUuMZ8Ml\nBOxw97Ow2H5S64QxUecfPXXThvzej6s3rkENh/bQcD7ESnQx0nXiyDSI3TV4iDqrqYqUUhVZKMLR\nyLFvugA+HAEtwci0lxBYqeF3d6Go61l0oXvlRIMG6ZL1RPY8mL6BXuuIbCt68a4SLunNI3WxTIei\nWaCAb23K3E3M4oopWsQoUzKvxMgOWH1dgX+NTVKO/S5mdGKWdQQ5E4pmaxuba8Q2Ef2YKknE/e0+\nBjhgHe4VKE83YH8ZrsTP0qqyNdbyjS5IzgXEf9XN6EOOSZJEWolMbPk8ydrS79REa3lA6RXYD4Dv\nOm+1CW9QauMY5YKYvJiFUEi/hDWmdDr5aGLAVkrkEsrP5mUQab4aR4a+EDAlOv9iIH68Ey6UKthQ\nLohJnJC3t07pFAkE8c5qkUudjwOUHdze8QvWT+5tlBI9PHmWjbT8shG0bOLEdJ08ik/pdHw1IvRf\nJ3g5yDsD+kisAP69B3zXOdTEJ+p5ZV8mXGIuiEmcKDVpaGjRpnTaRD8Fko0RwT/NxkXc1EQHMuj0\naQXy85CrQeHqto00iIRLzAUxiROjNfTGSemMS/RjIsHGupML6KQm9mlmAzNJK2DOND6G3yVyQUzi\nRHbSAT2FvSCa6OfI4BMZE2HZYI7UoHMkVoBFGv8eGCVymMSJiTnHIYp0TKKfI4NfzJiIp8CRHHT6\nvQI20iLhkpv3jiRGOi7RzyLwU3om5CLxiVcWMKD6uQIsLwETLp3CX1vjAL46jUv0Y5A7YUeZi5RC\nNjggj+wKiL8AkuTCWj6QkOjHRf655QonF3ChwkcH9P/vCuzo0bVHesQP4P1agelWb5a29QYfoPu2\nAj0maLJHgL65PDC0vBXYvjyYQjmpiT1pGoAP9wrc05OB3tA9mR6Au1yBnhI0vdTELk0PxPu6Aj0l\naHqpiX11fGCs2xXoJUFzXbfGBvJHcAV6SNCUqYlH0PeBaVyB/wH+hLUHWq5gZgAAAABJRU5ErkJg\ngg==\n", "text": [ "D_1zn__(l)\u22c5m_1__(l)\u22c5(a_n__(l+1)\u22c5\u03b6_z1n__(l) - d_n__(l+1)\u22c5\u03c8_z1n__(l)) - m__(l)\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\u2500\u2500\u2500\n", " m_1__(l)\u22c5\u03b6_zn__(l)\u22c5(D_1zn\n", "\n", "-D_1z1n__(l)\u22c5d_n__(l+1)\u22c5\u03c8_z1n__(l) + D_3z1n__(l)\u22c5a_n__(l+1)\u22c5\u03b6_z1n__(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\n", "__(l) - D_3zn__(l)) " ] }, { "latex": [ "Result for $d^{(l)}_n$:" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "latex": [ "$$\\frac{1}{m^{(l)}_{1} \\psi^{(l)}_{zn} \\left(D^{(l)}_{1zn} - D^{(l)}_{3zn}\\right)} \\left(D^{(l)}_{3zn} m^{(l)}_{1} \\left(a^{(l+1)}_{n} \\zeta^{(l)}_{z1n} - d^{(l+1)}_{n} \\psi^{(l)}_{z1n}\\right) - m^{(l)} \\left(- D^{(l)}_{1z1n} d^{(l+1)}_{n} \\psi^{(l)}_{z1n} + D^{(l)}_{3z1n} a^{(l+1)}_{n} \\zeta^{(l)}_{z1n}\\right)\\right)$$" ], "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAA9YAAABABAMAAAANEPJCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnCHksXj4j8Dw6Z82pqHyMy1qLKzvCPwfPYXLLlLub1MAAAAASUVORK5CYII=\n", "text": [ "D_3zn__(l)\u22c5m_1__(l)\u22c5(a_n__(l+1)\u22c5\u03b6_z1n__(l) - d_n__(l+1)\u22c5\u03c8_z1n__(l)) - m__(l)\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\u2500\u2500\u2500\n", " m_1__(l)\u22c5\u03c8_zn__(l)\u22c5(D_1zn\n", "\n", "-D_1z1n__(l)\u22c5d_n__(l+1)\u22c5\u03c8_z1n__(l) + D_3z1n__(l)\u22c5a_n__(l+1)\u22c5\u03b6_z1n__(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\n", "__(l) - D_3zn__(l)) " ] }, { "latex": [ "Result for $b^{(l)}_n$:" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "latex": [ "$$\\frac{1}{m^{(l)}_{1} \\zeta^{(l)}_{zn} \\left(D^{(l)}_{1zn} - D^{(l)}_{3zn}\\right)} \\left(D^{(l)}_{1zn} m^{(l)} \\left(b^{(l+1)}_{n} \\zeta^{(l)}_{z1n} - c^{(l+1)}_{n} \\psi^{(l)}_{z1n}\\right) - m^{(l)}_{1} \\left(- D^{(l)}_{1z1n} c^{(l+1)}_{n} \\psi^{(l)}_{z1n} + D^{(l)}_{3z1n} b^{(l+1)}_{n} \\zeta^{(l)}_{z1n}\\right)\\right)$$" ], "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAA8oAAABABAMAAAAwgpMkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSh1GeAjn/n9P/V9dsKyxOHu4SwAAAABJRU5ErkJggg==\n", "text": [ "D_1zn__(l)\u22c5m__(l)\u22c5(b_n__(l+1)\u22c5\u03b6_z1n__(l) - c_n__(l+1)\u22c5\u03c8_z1n__(l)) - m_1__(l)\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\u2500\u2500\u2500\n", " m_1__(l)\u22c5\u03b6_zn__(l)\u22c5(D_1zn\n", "\n", "-D_1z1n__(l)\u22c5c_n__(l+1)\u22c5\u03c8_z1n__(l) + D_3z1n__(l)\u22c5b_n__(l+1)\u22c5\u03b6_z1n__(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\n", "__(l) - D_3zn__(l)) " ] }, { "latex": [ "Result for $c^{(l)}_n$:" ], "metadata": {}, "output_type": "display_data", "text": [ "" ] }, { "latex": [ "$$\\frac{1}{m^{(l)}_{1} \\psi^{(l)}_{zn} \\left(D^{(l)}_{1zn} - D^{(l)}_{3zn}\\right)} \\left(D^{(l)}_{3zn} m^{(l)} \\left(b^{(l+1)}_{n} \\zeta^{(l)}_{z1n} - c^{(l+1)}_{n} \\psi^{(l)}_{z1n}\\right) - m^{(l)}_{1} \\left(- D^{(l)}_{1z1n} c^{(l+1)}_{n} \\psi^{(l)}_{z1n} + D^{(l)}_{3z1n} b^{(l+1)}_{n} \\zeta^{(l)}_{z1n}\\right)\\right)$$" ], "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAA84AAABABAMAAAA5aTNeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYCGKBk3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"text": [ "D_3zn__(l)\u22c5m__(l)\u22c5(b_n__(l+1)\u22c5\u03b6_z1n__(l) - c_n__(l+1)\u22c5\u03c8_z1n__(l)) - m_1__(l)\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\u2500\u2500\u2500\n", " m_1__(l)\u22c5\u03c8_zn__(l)\u22c5(D_1zn\n", "\n", "-D_1z1n__(l)\u22c5c_n__(l+1)\u22c5\u03c8_z1n__(l) + D_3z1n__(l)\u22c5b_n__(l+1)\u22c5\u03b6_z1n__(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\n", "__(l) - D_3zn__(l)) " ] } ], "prompt_number": 14 }, { "cell_type": "code", "collapsed": false, "input": [ "single = all_eq[0]\n", "single = single * m__l_1*zeta__l_zn*(D__l_1zn-D__l_3zn)\n", "single = simplify(single.replace(d__lp1_n, 1).replace(m__l_1, 1).replace(m__l, 1))\n", "\n", "display(single)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$D^{(l)}_{1z1n} \\psi^{(l)}_{z1n} + D^{(l)}_{1zn} \\left(a^{(l+1)}_{n} \\zeta^{(l)}_{z1n} - \\psi^{(l)}_{z1n}\\right) - D^{(l)}_{3z1n} a^{(l+1)}_{n} \\zeta^{(l)}_{z1n}$$" ], "metadata": {}, "output_type": "display_data", "png": 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"text": [ "D_1z1n__(l)\u22c5\u03c8_z1n__(l) + D_1zn__(l)\u22c5(a_n__(l+1)\u22c5\u03b6_z1n__(l) - \u03c8_z1n__(l)) - D_3\n", "z1n__(l)\u22c5a_n__(l+1)\u22c5\u03b6_z1n__(l)" ] } ], "prompt_number": 23 }, { "cell_type": "code", "collapsed": false, "input": [ "z_nm1, z_np1, z_n, n, rho = symbols(\"z_n-1, z_n+1, z_n, n, rho\")\n", "z_np1 = ((2*n+1)/rho)*z_n - z_nm1\n", "z__dash_n = (n*z_nm1 - (n+1)*z_np1)/(2*n+1)\n", "display(collect(collect(expand(simplify(z__dash_n*rho+z_n)),z_n),rho))\n", "#display(z_np1)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- n z_{n} + \\rho z_{n-1}$$" ], "metadata": {}, "output_type": "display_data", "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAQBAMAAADJxG2SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n", "text": [ "-n\u22c5z_n + \u03c1\u22c5z_n-1" ] } ], "prompt_number": 46 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }