数学 - Python - JavaScript - 解析学 - 多変数の関数 - 積分記号下の微分(応用例、帰納法)

解析入門〈3〉

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.5(積分記号下の微分)、問題2.を取り組んでみる。

2. 
   

   左辺について。

   $\frac{d}{da}\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-a{x}^2\right)}\mathrm{dx}=\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-a{x}^2\right)}\left(-x^2\right)\mathrm{dx}$

   また、

   $\begin{array}{}\frac{d^n}{d{a}^n}\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-a{x}^2\right)}\mathrm{dx}\\ =\frac{d}{da}\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-a{x}^2\right)}{\left(-1\right)}^{\left(n-1\right)}{x}^{\left(2\left(n-1\right)\right)}\mathrm{dx}\\ ={\int \; <!--\; integral\; -->}_0^{\mathrm{\infty \; <!--\; infinity\; -->}}{e}^{\left(-a{x}^2\right)}{\left(-1\right)}^n{x}^{\left(2n\right)}\mathrm{dx}\end{array}$

   右辺について。

   $\begin{array}{}\frac{d}{da}\sqrt{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{a}}\\ =\frac{d}{da}{\left(\frac{a}{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}\right)}^{\left(-\frac{1}{2}\right)}\\ =-\frac{\sqrt{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}}{2}{a}^{\left(-\frac{3}{2}\right)}\end{array}$

   また、

   $\begin{array}{}\frac{d^n}{d{a}^n}\sqrt{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{a}}\\ ={\left(-1\right)}^{n-1}\frac{d}{da}\left(\frac{1·\; <!--\; middle\; dot\; -->3·\; <!--\; middle\; dot\; -->5\dots \; <!--\; horizontal\; ellipsis\; -->\left(2n-3\right)}{2^n}\right)\sqrt{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{a}^{\left(-\frac{2n-1}{2}\right)}\\ ={\left(-1\right)}^n\frac{1·\; <!--\; middle\; dot\; -->3·\; <!--\; middle\; dot\; -->5\dots \; <!--\; horizontal\; ellipsis\; -->\left(2n-3\right)\left(2n-1\right)}{2^{\left(n+1\right)}}\sqrt{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{a}^{\left(-\frac{2n-3}{2}\right)}\end{array}$

   よって、帰納法より、

   ${\left(-1\right)}^n\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-a{x}^2\right)}{x}^{\left(2n\right)}\mathrm{dx}={\left(-1\right)}^n\frac{1·\; <!--\; middle\; dot\; -->3·\; <!--\; middle\; dot\; -->5\dots \; <!--\; horizontal\; ellipsis\; -->\left(2n-1\right)}{2^{\left(n+1\right)}}\sqrt{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{a}^{\left(-\frac{2n-3}{2}\right)}$

   a に1を代入すると、

   $\underset{0}{\overset{\mathrm{\infty \; <!--\; infinity\; -->}}{\int \; <!--\; integral\; -->}}{e}^{\left(-x^2\right)}{x}^{\left(2n\right)}\mathrm{dx}=\frac{1·\; <!--\; middle\; dot\; -->3·\; <!--\; middle\; dot\; -->5\dots \; <!--\; horizontal\; ellipsis\; -->\left(2n-1\right)}{2^{\left(n+1\right)}}\sqrt{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, exp, oo, pi, sqrt, plot, Rational

a = symbols('a', positive=True)
n = symbols('n', integer=True)
x = symbols('x')
f = exp(-a * x ** 2) * x ** (2 * n)
I = Integral(f, (x, 0, oo))
d = {a: 1, n: 10}
for t in [I, I.doit(), I.subs(d).doit()]:
    pprint(t)
    print()

result = 1
for i in range(1, 11):
    result *= (2 * i - 1)
pprint(Rational(result, 2 ** (10 + 1)) * sqrt(pi))

p = plot(f.subs(d), show=False, legend=True)
p.save('sample2.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample2.py
∞               
⌠               
⎮           2   
⎮  2⋅n  -a⋅x    
⎮ x   ⋅ℯ      dx
⌡               
0               

⎧ -n + 1/2                             
⎪a        ⋅Γ(n + 1/2)                  
⎪────────────────────  for -n + 1/2 < 1
⎪        2⋅a                           
⎪                                      
⎪  ∞                                   
⎨  ⌠                                   
⎪  ⎮           2                       
⎪  ⎮  2⋅n  -a⋅x                        
⎪  ⎮ x   ⋅ℯ      dx       otherwise    
⎪  ⌡                                   
⎪  0                                   
⎩                                      

654729075⋅√π
────────────
    2048    

654729075⋅√π
────────────
    2048    
$

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="0.005">
<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="n0">n = </label>
<input id="n0" type="number" value="2">

<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="sample2.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,
              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 fns = [[(x) => Math.sin(x) ** 2, 'red']];

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),
        n0 = parseInt(input_n0.value, 10);
            
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        fns = [[(x) => Math.exp(-(x ** 2)) * x ** (2 * n0), 'red']],
        lines = [];
    
    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();

(x) => Math.exp(-(x ** 2)) * x ** (2 * n0),red

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