Juliaで複素積分

ComplexIntegralJulia.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Juliaで複素積分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 複素積分∮の定義"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "∮ (generic function with 1 method)"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ∮(f,path)\n",
    "    s=0\n",
    "    for i in 2:length(path)\n",
    "        s=s+f((path[i]+path[i-1])/2)*(path[i]-path[i-1])\n",
    "    end\n",
    "    return s\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 複素平面上での四角い積分経路"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "□ (generic function with 4 methods)"
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function □(leftdown,rightup,num)\n",
    "    r1=real(leftdown)\n",
    "    r2=real(rightup)\n",
    "    i1=imag(leftdown)\n",
    "    i2=imag(rightup)\n",
    "    a=linspace(r1,r2,num)+i1*im\n",
    "    b=linspace(i1,i2,num)*im+r2\n",
    "    c=linspace(r2,r1,num)+i2*im\n",
    "    d=linspace(i2,i1,num)*im+r1\n",
    "    return vcat(vcat(a,b),vcat(c,d))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 複素関数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(z)=z*3\n",
    "f(z)=1/z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "gは極を持たない複素平面全域で正則な関数\n",
    "\n",
    "fは原点に極を持つ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 積分の実行"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 極を囲わない積分路"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.881784197001252e-16 + 3.552713678800501e-15im"
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cc=□(3+im,4+3*im,4)\n",
    "∮(g,cc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0 + 0.0im"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "∮(g,□(3+im,0+im,4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5543122344752192e-15 - 4.440892098500626e-16im"
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cr=□(1+im,3+im)\n",
    "∮(g,cr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0547118733938987e-15 - 3.552713678800501e-15im\n",
      "-0.00018306775067675196 + 0.0009021823204503621im\n"
     ]
    }
   ],
   "source": [
    "c=□(1+im,3+2*im)\n",
    "println(∮(g,c))\n",
    "println(∮(f,c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### fの極を囲う積分路"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0 + 0.0im\n",
      "9.71445146547012e-17 + 6.291421562431563im\n"
     ]
    }
   ],
   "source": [
    "c=□(-5+-5im,4+4*im)\n",
    "println(∮(g,c))\n",
    "println(∮(f,c))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 251,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.283185307179586"
      ]
     },
     "execution_count": 251,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2*π"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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