数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - スターリングの公式(定理の証明1-3、微分(合成関数、連鎖律、対数関数、累乗、分数関数))

解析入門 原書第3版
原著
楽天ブックス(Kobo)

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 - MyScript(iPad アプリ(iOS))
• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス(Kobo))
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo))
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo))

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12章(いくつかの計算練習)、2(スターリングの公式)の定理2の証明1、2、3.を取り組んでみる。

1. 
   
   $\begin{array}{}\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\text{'}\left(x\right)\\ =\frac{d}{\mathrm{dx}}\left(\frac{1}{2}\log \frac{1+x}{1-x}-x\right)\\ =\frac{1}{2}·\; <!--\; middle\; dot\; -->\frac{1-x}{1+x}·\; <!--\; middle\; dot\; -->\frac{\left(1-x\right)-\left(1+x\right)\left(-1\right)}{{\left(1-x\right)}^2}-1\\ =\frac{1}{2}·\; <!--\; middle\; dot\; -->\frac{1}{1+x}·\; <!--\; middle\; dot\; -->\frac{2}{1-x}-1\\ =\frac{1}{1-x^2}-1\\ =\frac{1-\left(1-x^2\right)}{1-x^2}\\ =\frac{x^2}{1-x^2}\end{array}$
2. 
   
   $\begin{array}{}\mathrm{\psi \; <!--\; greek\; small\; letter\; psi\; -->}\text{'}\left(x\right)\\ =\frac{d}{\mathrm{dx}}\left(\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\left(x\right)-\frac{x^3}{3\left(1-x^2\right)}\right)\\ =\frac{x^2}{1-x^2}-\frac{1}{3}·\; <!--\; middle\; dot\; -->\frac{3x^2\left(1-x^2\right)-x^3\left(-2x\right)}{{\left(1-x^2\right)}^2}\\ =\frac{x^2}{1-x^2}-\frac{1}{3}·\; <!--\; middle\; dot\; -->\frac{3x^2-x^4}{{\left(1-x^2\right)}^2}\\ =\frac{x^2·\; <!--\; middle\; dot\; -->3\left(1-x^2\right)-3x^2+x^4}{3{\left(1-x^2\right)}^2}\\ =\frac{-2x^4}{3{\left(1-x^2\right)}^2}\end{array}$
3. 
   
   $\begin{array}{}0<x<1\\ \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\left(0\right)=\frac{1}{2}\log 1=0\\ \mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\text{'}\left(x\right)>0\end{array}$

   よって、

   $\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\left(x\right)>0\left(0<x<1\right)$

   また、

   $\begin{array}{}0<x<1\\ \mathrm{\psi \; <!--\; greek\; small\; letter\; psi\; -->}\left(0\right)\\ =\mathrm{\phi \; <!--\; greek\; small\; letter\; phi\; -->}\left(0\right)\\ =0\\ \mathrm{\psi \; <!--\; greek\; small\; letter\; psi\; -->}\text{'}\left(x\right)<0\end{array}$

   よって、

   $\mathrm{\psi \; <!--\; greek\; small\; letter\; psi\; -->}\left(x\right)<0\left(0<x<1\right)$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, log, Derivative, plot

print('1.')

x = symbols('x')
f = Rational(1, 2) * log((1 + x) / (1 - x)) - x
f1 = Derivative(f, x, 1).doit()
pprint(f1.simplify())

print('2.')
g = f - x ** 3 / (3 * (1 - x ** 2))
g1 = Derivative(g, x, 1).doit()

pprint(g1.factor())

print('3.')

p = plot(f, g, (x, 0, 1), ylim=(-10, 10), show=False, legend=True)
colors = ['red', 'green']

for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample1.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
1.
   2  
 -x   
──────
 2    
x  - 1
2.
           4       
       -2⋅x        
───────────────────
         2        2
3⋅(x - 1) ⋅(x + 1) 
3.
$

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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="2">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<label for="y2">y2 = </label>
<input id="y2" 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="sample1.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 / 2 * Math.log((1 + x) / (1 - x)) - x,
    fns = [[f, 'red'],
           [(x) => f(x) - x ** 3 / (3 * (1 - x ** 2)), 'green']];

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 = [],
        lines = [[0, y1, 0, y2, 'blue'],
                 [1, y1, 1, y2, 'brown']];
    
    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 =