{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "今回扱うガウスの超幾何関数$F(a,b,c,z)$は次のセルの式で定義されます。ここで登場する$(a)_n$はポッホハマー記号と呼ばれ次の式で定義されます。\n",
    "$$(a)_n=\\prod_{k=0}^{n-1}(a+k)=\\frac{\\Gamma(a+n)}{\\Gamma(a)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{1}$}F\\left( \\left. \\begin{array}{c}a,\\;b\\\\c\\end{array} \\right |,z\\right)=\\sum_{n=0}^{\\infty }{\\frac{\\left(a\\right)_{n}\\,\\left(b\\right)_{n}\\,z^{n}}{\\left(c\\right)_{n}\\,n!}}\\]"
      ],
      "text/plain": [
       "                                               inf              n\n",
       "                                               ====  (a)  (b)  z\n",
       "                                               \\        n    n\n",
       "(%o1)         hypergeometric([a, b], [c], z) =  >    ------------\n",
       "                                               /       (c)  n!\n",
       "                                               ====       n\n",
       "                                               n = 0"
      ],
      "text/x-maxima": [
       "hypergeometric([a,b],[c],z) = 'sum((pochhammer(a,n)*pochhammer(b,n)*z^n)\n",
       "                                    /(pochhammer(c,n)*n!),n,0,inf)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-HYPERGEOMETRIC in DEFUN\n",
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-UNIT-STEP in DEFUN\n",
      "SB-KERNEL:REDEFINITION-WITH-DEFUN: redefining MAXIMA::SIMP-POCHHAMMER in DEFUN\n"
     ]
    }
   ],
   "source": [
    "hypergeometric([a,b],[c],z)=sum(pochhammer(a,n)*pochhammer(b,n)/(pochhammer(c,n)*n!)*z^n,n,0,inf);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "超幾何関数は$a,b,c,z$に様々な値や式を代入することで、多くの初等関数や特殊関数を表すことができます。ここでは$\\cos{z}$が次の超幾何関数で表されることを見てます。確認のために次の超幾何関数と$\\cos{z}$をテイラー展開して係数が一致することを見てみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{2}$}F\\left( \\left. \\begin{array}{c}\\\\\\frac{1}{2}\\end{array} \\right |,-\\frac{z^2}{4}\\right)\\]"
      ],
      "text/plain": [
       "                                                    2\n",
       "                                             1     z\n",
       "(%o2)                    hypergeometric([], [-], - --)\n",
       "                                             2     4"
      ],
      "text/x-maxima": [
       "hypergeometric([],[1/2],-z^2/4)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hypergeometric([],[1/2],-z^2/4);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{3}$}-\\frac{z^{14}}{87178291200}+\\frac{z^{12}}{479001600}-\\frac{z^{10}}{3628800}+\\frac{z^8}{40320}-\\frac{z^6}{720}+\\frac{z^4}{24}-\\frac{z^2}{2}+1\\]"
      ],
      "text/plain": [
       "               14            12         10       8      6     4    2\n",
       "              z             z          z        z      z     z    z\n",
       "(%o3)  (- -----------) + --------- - ------- + ----- - --- + -- - -- + 1\n",
       "          87178291200    479001600   3628800   40320   720   24   2"
      ],
      "text/x-maxima": [
       "(-z^14/87178291200)+z^12/479001600-z^10/3628800+z^8/40320-z^6/720+z^4/24-z^2/2\n",
       "                   +1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(%o2,z,0,15);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{4}$}-\\frac{z^{14}}{87178291200}+\\frac{z^{12}}{479001600}-\\frac{z^{10}}{3628800}+\\frac{z^8}{40320}-\\frac{z^6}{720}+\\frac{z^4}{24}-\\frac{z^2}{2}+1\\]"
      ],
      "text/plain": [
       "               14            12         10       8      6     4    2\n",
       "              z             z          z        z      z     z    z\n",
       "(%o4)  (- -----------) + --------- - ------- + ----- - --- + -- - -- + 1\n",
       "          87178291200    479001600   3628800   40320   720   24   2"
      ],
      "text/x-maxima": [
       "(-z^14/87178291200)+z^12/479001600-z^10/3628800+z^8/40320-z^6/720+z^4/24-z^2/2\n",
       "                   +1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylor(cos(z),z,0,15);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "超幾何関数が初等関数や特殊関数で表せる場合に、Maximaではそのような簡約を実行することができます。%o2として定義した超幾何関数を簡約してみましょう。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{6}$}\\cos z\\]"
      ],
      "text/plain": [
       "(%o6)                               cos(z)"
      ],
      "text/x-maxima": [
       "cos(z)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "besselexpand:true$\n",
    "hypergeometric_simp(%o2);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{7}$}F\\left( \\left. \\begin{array}{c}\\\\\\frac{1}{2}\\end{array} \\right |,-\\frac{z^2}{4}\\right)\\]"
      ],
      "text/plain": [
       "                                                    2\n",
       "                                             1     z\n",
       "(%o7)                    hypergeometric([], [-], - --)\n",
       "                                             2     4"
      ],
      "text/x-maxima": [
       "hypergeometric([],[1/2],-z^2/4)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hypergeometric([],[1/2],-z^2/4);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上記の式で定義された超幾何関数が$\\cos{z}$に等しいことを級数展開の係数が一致することで証明します。\n",
    "$$\\cos{z}=\\sum_{n=0}^{\\infty}\\frac{(-1)^n\\,z^{2\\,n}}{(2\\,n)!}$$\n",
    "は既知とします。\n",
    "超幾何関数の定義式より、$F\\left( \\left. \\begin{array}{c}\\\\c\\end{array} \\right |,x\\right)=\\sum_{n=0}^{\\infty}\\frac{x^n}{(c)_n\\,n!}$ですから、この総和記号の中身と$\\cos{z}$の展開式の総和記号の中身が一致することを示します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{9}$}\\frac{x^{n}}{\\left(c\\right)_{n}\\,n!}\\]"
      ],
      "text/plain": [
       "                                       n\n",
       "                                      x\n",
       "(%o9)                               -------\n",
       "                                    (c)  n!\n",
       "                                       n"
      ],
      "text/x-maxima": [
       "x^n/(pochhammer(c,n)*n!)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "declare(n,integer)$\n",
    "1/pochhammer(c,n)*x^n/n!;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ポッホハマー記号および階乗記号をガンマ関数に書き直してみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{10}$}\\frac{\\Gamma\\left(c\\right)\\,x^{n}}{\\Gamma\\left(n+1\\right)\\,\\Gamma\\left(n+c\\right)}\\]"
      ],
      "text/plain": [
       "                                            n\n",
       "                                  gamma(c) x\n",
       "(%o10)                     -------------------------\n",
       "                           gamma(n + 1) gamma(n + c)"
      ],
      "text/x-maxima": [
       "(gamma(c)*x^n)/(gamma(n+1)*gamma(n+c))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "makegamma(%);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ここに$c=\\frac{1}{2}, x=-\\frac{z^2}{4}$を代入してみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{11}$}\\frac{\\sqrt{\\pi}\\,\\left(-1\\right)^{n}\\,z^{2\\,n}}{4^{n}\\,\\Gamma\\left(n+\\frac{1}{2}\\right)\\,\\Gamma\\left(n+1\\right)}\\]"
      ],
      "text/plain": [
       "                                           n  2 n\n",
       "                            sqrt(%pi) (- 1)  z\n",
       "(%o11)                   ----------------------------\n",
       "                          n           1\n",
       "                         4  gamma(n + -) gamma(n + 1)\n",
       "                                      2"
      ],
      "text/x-maxima": [
       "(sqrt(%pi)*(-1)^n*z^(2*n))/(4^n*gamma(n+1/2)*gamma(n+1))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CTERM:%,c=1/2,x=-z^2/4;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "半整数に対するガンマ関数の計算をルールとして定義してそれを適用します。定義しているルールは、\n",
    "$$\\Gamma(n+\\frac{1}{2})=\\sqrt{\\pi}\\,\\frac{(2\\,n-1)!!}{2^n}$$\n",
    "です。ただ、Maximaの2重階乗の処理の都合でgenfact関数に変換されて表示されます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{13}$}{\\it gamma\\_half\\_int}:\\Gamma\\left(\\exp +\\frac{1}{2}\\right)\\rightarrow \\frac{\\sqrt{\\pi}\\,{\\it genfact}\\left(2\\,\\exp -1 , \\frac{2\\,\\exp -1}{2} , 2\\right)}{2^{\\exp }}\\]"
      ],
      "text/plain": [
       "                                    1\n",
       "(%o13) gamma_half_int : gamma(exp + -) -> \n",
       "                                    2\n",
       "                                                                  2 exp - 1\n",
       "                                     sqrt(%pi) genfact(2 exp - 1, ---------, 2)\n",
       "                                                                      2\n",
       "                                     ------------------------------------------\n",
       "                                                         exp\n",
       "                                                        2"
      ],
      "text/x-maxima": [
       "gamma_half_int:marrow(gamma(exp+1/2),\n",
       "                      (sqrt(%pi)*''genfact(2*exp-1,(2*exp-1)/2,2))/2^exp)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matchdeclare(exp,lambda([exp],featurep(exp,integer)))$\n",
    "defrule(gamma_half_int,gamma(exp+1/2),sqrt(%pi)*(2*exp-1)!!/2^exp);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{14}$}\\frac{\\left(-1\\right)^{n}\\,2^{n}\\,z^{2\\,n}}{4^{n}\\,\\Gamma\\left(n+1\\right)\\,{\\it genfact}\\left(2\\,n-1 , \\frac{2\\,n-1}{2} , 2\\right)}\\]"
      ],
      "text/plain": [
       "                                     n  n  2 n\n",
       "                                (- 1)  2  z\n",
       "(%o14)           --------------------------------------------\n",
       "                  n                               2 n - 1\n",
       "                 4  gamma(n + 1) genfact(2 n - 1, -------, 2)\n",
       "                                                     2"
      ],
      "text/x-maxima": [
       "((-1)^n*2^n*z^(2*n))/(4^n*gamma(n+1)*genfact(2*n-1,(2*n-1)/2,2))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apply1(CTERM,gamma_half_int);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[\\tag{${\\it \\%o}_{15}$}\\frac{\\left(-1\\right)^{n}\\,z^{2\\,n}}{2^{n}\\,\\Gamma\\left(n+1\\right)\\,{\\it genfact}\\left(2\\,n-1 , \\frac{2\\,n-1}{2} , 2\\right)}\\]"
      ],
      "text/plain": [
       "                                      n  2 n\n",
       "                                 (- 1)  z\n",
       "(%o15)           --------------------------------------------\n",
       "                  n                               2 n - 1\n",
       "                 2  gamma(n + 1) genfact(2 n - 1, -------, 2)\n",
       "                                                     2"
      ],
      "text/x-maxima": [
       "((-1)^n*z^(2*n))/(2^n*gamma(n+1)*genfact(2*n-1,(2*n-1)/2,2))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%,radcan;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "色々と約分されてようやくここまで変形できました。$genfact(2\\,n-1,\\frac{(2\\,n-1)}{2},2)=(2\\,n-1)!!$であることを思い出してください。これは$2\\,n-1$以下の全ての奇数の積です。\n",
    "\n",
    "$\\Gamma(n+1)=n!$なのでこれは$n$以下の全ての自然数の積です。最初の$2^n$をこの$n!$に分けてかけると、$2\\,n$以下の全ての偶数の積が得られます。つまり$2^n\\cdot \\Gamma(n+1)=(2\\,n)!!$です。\n",
    "\n",
    "したがって上記の式の分母は$(2\\,n)!!\\cdot(2\\,n-1)!!$となるのですが、この積は結局$2\\,n$以下の全ての自然数の積になります。つまり$(2\\,n)!$に等しくなります。というわけで上記の式は次の式に等しくなります。\n",
    "$$\\frac{\\left(-1\\right)^{n}\\,z^{2\\,n}}{(2\\,n)!}$$\n",
    "\n",
    "$\\cos{z}$のテイラー展開と一致していることが分かります。\n",
    "（証明終了）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 超幾何関数の定義とcos zの超幾何関数表現\n",
    "\n",
    "Copyright 2021 Yasuaki Honda (yasuaki.honda@gmail.com)\n",
    "GPL2\n",
    "\n",
    "If you want to use this file with different licensing terms, please email me."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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