数学 - Python - JavaScript - 解析学 - 積分法 - 不定積分の計算(漸化式、三角関数、正弦、余弦)

解析入門〈2〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.1(不定積分の計算)、問題4.を取り組んでみる。

4. 
   $\begin{array}{l}\left(m+1\right)I\left(m,n\right)\\ =\left(m+1\right){\displaystyle \int {\sin }^{m}x{\cos }^{n}xdx}\\ ={\displaystyle \int \left(m+1\right){\sin }^{m}x{\cos }^{n}xdx}\\ ={\displaystyle \int \left(m+1\right){\sin }^{m}x\cos x{\cos }^{n-1}xdx}\\ ={\displaystyle \int \frac{d}{dx}\left({\sin }^{m+1}x\right){\cos }^{n-1}xdx}\\ ={\sin }^{m+1}x{\cos }^{n-1}x-{\displaystyle \int {\sin }^{m+1}x\left(n-1\right){\cos }^{n-2}x\left(-\sin x\right)dx}\\ ={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right){\displaystyle \int {\sin }^{m+2}x{\cos }^{n-2}xdx}\\ ={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right){\displaystyle \int {\sin }^{m}x{\cos }^{n-2}x{\sin }^{2}xdx}\\ ={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right){\displaystyle \int {\sin }^{m}x{\cos }^{n-2}x\left(1-{\cos }^{2}x\right)xdx}\\ ={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right)\left({\displaystyle \int {\sin }^{m}x{\cos }^{n-2}xdx-{\displaystyle \int {\sin }^{m}x{\cos }^{n}xdx}}\right)\\ ={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right)\left(I\left(m,n-2\right)-I\left(m,n\right)\right)\\ \\ \left(m+1\right)I\left(m,n\right)={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right)\left(I\left(m,n-2\right)-I\left(m,n\right)\right)\\ \left(m+n\right)I\left(m,n\right)={\sin }^{m+1}x{\cos }^{n-1}+\left(n-1\right)I\left(m,n-2\right)\\ \\ \left(n+1\right)I\left(m,n\right)\\ ={\displaystyle \int \left(n+1\right){\sin }^{m}x{\cos }^{n}xdx}\\ ={\displaystyle \int {\sin }^{m-1}x\frac{d}{dx}\left(-{\cos }^{n+1}x\right)dx}\\ =-{\sin }^{m-1}{\cos }^{n+1}x+{\displaystyle \int \left(m-1\right){\sin }^{m-2}x{\cos }^{n+2}xdx}\\ =-{\sin }^{m-1}x{\cos }^{n+1}x+\left(m-1\right){\displaystyle \int {\sin }^{m-2}x{\cos }^{n}x{\cos }^{2}xdx}\\ =-{\sin }^{m-1}x{\cos }^{n+1}x+\left(m-1\right){\displaystyle \int {\sin }^{m-2}x{\cos }^{n}x\left(1-{\sin }^{2}x\right)dx}\\ =-{\sin }^{m-1}x{\cos }^{n+1}x+\left(m-1\right)\left(I\left(m-2,n\right)-I\left(m,n\right)\right)\\ \\ \left(n+1\right)I\left(m,n\right)=-{\sin }^{m-1}x{\cos }^{n+1}x+\left(m-1\right)\left(I\left(m-2,n\right)-I\left(m,n\right)\right)\\ \left(m+n\right)I\left(m,n\right)=-{\sin }^{m-1}x{\cos }^{n+1}x+\left(m-1\right)I\left(m-2,n\right)\end{array}$
   1. 
      $\begin{array}{l}m+n\ne 0\\ I\left(m,n\right)=\frac{{\sin }^{m+1}x{\cos }^{n-1}}{m+n}+\frac{n-1}{m+n}I\left(m,n-2\right)\\ I\left(m,n\right)=-\frac{{\sin }^{m-1}x{\cos }^{n+1}x}{m+n}+\frac{m-1}{m+n}I\left(m-2,n\right)\end{array}$
   2. 
      $\begin{array}{l}\left(m+n+2\right)I\left(m,n+2\right)={\sin }^{m+1}x{\cos }^{n+1}x+\left(n+1\right)I\left(m,n\right)\\ \left(n+1\right)I\left(m,n\right)=-{\sin }^{m+1}x{\cos }^{n+1}x+\left(m+n+2\right)I\left(m,n+2\right)\\ n\ne -1\\ I\left(m,n\right)=-\frac{{\sin }^{m+1}x{\cos }^{n+1}x}{n+1}+\frac{m+n+2}{n+1}I\left(m,n+2\right)\\ \\ I\left(m+2,n\right)=-\frac{{\sin }^{m+1}x{\cos }^{n+1}x}{m+n+2}+\frac{m+1}{m+n+2}I\left(m,n\right)\\ \left(m+1\right)I\left(m,n\right)={\sin }^{m+1}x{\cos }^{n+1}x+\left(m+n+2\right)I\left(m+2,n\right)\\ m\ne -1\\ I\left(m,n\right)=\frac{{\sin }^{m+1}x{\cos }^{n+1}x}{m+1}+\frac{m+n+2}{m+1}I\left(m+2,n\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, sin, cos, plot

print('4.')
m, n = symbols('m n', integer=True)
x = symbols('x')
I = Integral(sin(x) ** m + cos(x) ** n, x)
pprint(I)

m0 = 2
n0 = 3
eq = (I.subs({m: m0, n: n0}) - (
    sin(x) ** (m + 1) * cos(x) **
    (n - 1) / (m + n) + (n - 1) / (m + n) + I.subs(
        {m: m0, n: n0 - 2}))).subs({m: m0, n: n0})

pprint(eq)

入出力結果(Terminal, IPython)

$ ./sample4.py
4.
⌠                       
⎮ ⎛   m         n   ⎞   
⎮ ⎝sin (x) + cos (x)⎠ dx
⌡                       
     3       2      ⌠                         ⌠                           
  sin (x)⋅cos (x)   ⎮ ⎛   2            ⎞      ⎮ ⎛   2         3   ⎞      2
- ─────────────── - ⎮ ⎝sin (x) + cos(x)⎠ dx + ⎮ ⎝sin (x) + cos (x)⎠ dx - ─
         5          ⌡                         ⌡                          5
$         

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="m0">m0 = </label>
<input id="m0" type="number" step="1" value="2">
<label for="n0">n0 = </label>
<input id="n0" type="number" step="1" value="3">
     
<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample4.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_m0 = document.querySelector('#m0'),
    input_n0 = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_m0, input_n0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        m0 = parseFloat(input_m0.value),
        n0 = parseFloat(input_n0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [],
        f = (x) => Math.sin(x) ** m0 * Math.cos(x) ** n0,
        fns = [[f, 'green']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
m0 = n0 =