数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(連続関数、周期関数、円周率)

解析入門 原書第3版
原著

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 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)、3(三角関数の積分)、練習問題20.を取り組んでみる。

20. 
    
    $\begin{array}{}{\int \; <!--\; integral\; -->}_{a+2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{b+2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f\left(x\right)\mathrm{dx}\\ ={\int \; <!--\; integral\; -->}_a^{b}f\left(x\right)\mathrm{dx}\end{array}$
    $\begin{array}{}{\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c}f\\ ={\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c}^{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f+{\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f+{\int \; <!--\; integral\; -->}_{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c}f\\ ={\int \; <!--\; integral\; -->}_{\left(\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c\right)}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f+{\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f+{\int \; <!--\; integral\; -->}_{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c}f\\ ={\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f+{\int \; <!--\; integral\; -->}_{\left(\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c\right)}^{\left(\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+c\right)}f\\ ={\int \; <!--\; integral\; -->}_{-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}f\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, pi, sin, cos, Integral, Function, plot

x, a, b, c = symbols('x, a, b, c')

for f in [sin(x), cos(x)]:
    I1 = Integral(f, (x, a + 2 * pi, b + 2 * pi))
    I2 = Integral(f, (x, a, b))
    I3 = Integral(f, (x, -pi + c, pi + c))
    I4 = Integral(f, (x, -pi, pi))

    for t in [I1, I2, I1.doit(), I1.doit() == I2.doit(),
              I3, I4, I3.doit(), I3.doit() == I4.doit()]:
        pprint(t)
        print()

p = plot(sin(x), cos(x), legend=True, show=False)
for i, color in enumerate(['red', 'green']):
    p[i].line_color = color
p.save('sample26.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample26.py
b + 2⋅π          
   ⌠             
   ⎮    sin(x) dx
   ⌡             
a + 2⋅π          

b          
⌠          
⎮ sin(x) dx
⌡          
a          

cos(a) - cos(b)

True

c + π          
  ⌠            
  ⎮   sin(x) dx
  ⌡            
c - π          

π           
⌠           
⎮  sin(x) dx
⌡           
-π          

0

True

b + 2⋅π          
   ⌠             
   ⎮    cos(x) dx
   ⌡             
a + 2⋅π          

b          
⌠          
⎮ cos(x) dx
⌡          
a          

-sin(a) + sin(b)

True

c + π          
  ⌠            
  ⎮   cos(x) dx
  ⌡            
c - π          

π           
⌠           
⎮  cos(x) dx
⌡           
-π          

0

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="1" value="-1">
<label for="b0">b = </label>
<input id="b0" type="number" min="1" value="2">
<label for="c0">c = </label>
<input id="c0" type="number" min="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="sample26.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    input_c0 = document.querySelector('#c0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_a0, input_c0],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[(x) => Math.sin(x), 'red'],
           [(x) => Math.cos(x), '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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value),
        c0 = parseFloat(input_c0.value);
        

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [],
        lines = [[a0 + 2 * Math.PI, y1, a0 + 2 * Math.PI, y2, 'blue'],
                 [b0 + 2 * Math.PI, y1, b0 + 2 * Math.PI, y2, 'blue'],
                 [a0, y1, a0, y2, 'orange'],
                 [b0, y1, b0, y2, 'orange'],
                 [-Math.PI + c0, y1, -Math.PI + c0, y2, 'brown'],
                 [Math.PI + c0, y1, Math.PI + c0, y2, 'brown'],
                 [-Math.PI, y1, -Math.PI, y2, 'skyblue'],
                 [Math.PI, y1, Math.PI, y2, 'skyblue']];

    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 =
a = b = c =