数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - いろいろな問題(部分積分法、帰納法、階乗、指数関数、定積分)

解析入門 原書第3版
原著

学習環境

• Surface Go、タイプ カバー、ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ(iOS))
• 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 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第11章(積分の計算)、補充問題(いろいろな問題)6.を取り組んでみる。

6. 
   
   $\begin{array}{}\int \; <!--\; integral\; -->{\left(1-x^2\right)}^n\mathrm{dx}\\ =x{\left(1-x^2\right)}^n-\int \; <!--\; integral\; -->xn\left(-2x\right){\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-x^2\right)}^n-2n\int \; <!--\; integral\; -->\left(-x^2\right){\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-x^2\right)}^n-2n\int \; <!--\; integral\; -->\left(-1+1-x^2\right){\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-x^2\right)}^n+2n\int \; <!--\; integral\; -->{\left(1-x^2\right)}^{n-1}\mathrm{dx}-2n\int \; <!--\; integral\; -->{\left(1-x^2\right)}^n\mathrm{dx}\end{array}$

   よって、

   $\begin{array}{}\left(2n+1\right)\int \; <!--\; integral\; -->{\left(1-x^2\right)}^n\mathrm{dx}=x{\left(1-x^2\right)}^n+2n\int \; <!--\; integral\; -->{\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ \int \; <!--\; integral\; -->{\left(1-x^2\right)}^n\mathrm{dx}=\frac{1}{2n+1}x{\left(1-x^2\right)}^n+\frac{2n}{2n+1}\int \; <!--\; integral\; -->{\left(1-x^2\right)}^{n-1}\mathrm{dx}\end{array}$

   ゆえに、

   $\begin{array}{}{\int \; <!--\; integral\; -->}_0^1\left(1-x^2\right)\mathrm{dx}\\ =\frac{1}{2n+1}{\left[x{\left(1-x^2\right)}^n\right]}_0^1+\frac{2n}{2n+1}{\int \; <!--\; integral\; -->}_0^1{\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ =\frac{2n}{2n+1}{\int \; <!--\; integral\; -->}_0^1{\left(1-x^2\right)}^{n-1}\mathrm{dx}\\ =\frac{2n}{2n+1}·\; <!--\; middle\; dot\; -->\frac{2^{2\left(n-1\right)}{\left(\left(n-1\right)!\right)}^2}{\left(2\left(n-1\right)+1\right)!}\\ =\frac{n}{2n+1}·\; <!--\; middle\; dot\; -->\frac{2^{2n-1}{\left(\left(n-1\right)!\right)}^2}{\left(2n-1\right)!}\\ =\frac{n}{2n+1}·\; <!--\; middle\; dot\; -->\frac{2n}{2n}·\; <!--\; middle\; dot\; -->\frac{2^{2n-1}{\left(\left(n-1\right)!\right)}^2}{\left(2n-1\right)!}\\ =\frac{2^{2n}{\left(n!\right)}^2}{\left(2n+1\right)!}\end{array}$

   以上から、 帰納法により、すべての負でない整数に対して成り立つ。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, plot, factorial

print('6.')

x = symbols('x')
n = symbols('n', integer=True, nonegative=True)
f = (1 - x ** 2) ** n

I = Integral(f, (x, 0, 1))
for t in [I, I.doit()]:
    pprint(t)
    print()

g = 2 ** (2 * n) * factorial(n) ** 2 / factorial(2 * n + 1)
p = plot(f.subs({n: 2}), g.subs({n: 2}), legend=True, show=False)
colors = ['red', 'green']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample6.svg')

pprint(g)

for n0 in range(10):
    print(f'n = {n0}:', I.subs({n: n0}).doit() == g.subs({n: n0}))

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
1               
⌠               
⎮           n   
⎮ ⎛   2    ⎞    
⎮ ⎝- x  + 1⎠  dx
⌡               
0               

 ┌─  ⎛1/2, -n │  ⎞
 ├─  ⎜        │ 1⎟
2╵ 1 ⎝  3/2   │  ⎠

  2⋅n   2 
 2   ⋅n!  
──────────
(2⋅n + 1)!
n = 0: True
n = 1: True
n = 2: True
n = 3: True
n = 4: True
n = 5: True
n = 6: True
n = 7: True
n = 8: True
n = 9: True
$

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="0.001" 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">

<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="sample6.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => (1 - x ** 2) ** 2,
    fns = [[f, 'red']],
    factorial = (n) => {
        let result = 1;
        for (let i = 1; i <= n; i += 1) {
            result *= i;
        }
        return result;
    }

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);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [],
        n0 = 2,
        yn = 2 ** (2 * n0) * factorial(n0) ** 2 / factorial(2 * n0 + 1),
        lines = [[x1, yn, x2, yn, 'green'],
                 [0, y1, 0, y2, 'blue'],
                 [1, y1, 1, y2, 'orange']];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

    [fns].forEach((fs) => p(fs.join('\n')));
};

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

r = dx =
x1 = x2 =
y1 = y2 =