数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(正弦、余弦、加法定理)

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

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8.を取り組んでみる。

$\begin{array}{l} \frac{1}{2} (\cos (m - n) x - \cos (m + n) x) \\ = \frac{1}{2} (\cos (m x) \cos (n x) + \sin (m x) \sin (n x) - \cos (m x) \cos (n x) + \sin (n x) \sin (n x)) \\ = \sin (m x) \sin (n x) \end{array}$

$\begin{array}{l} \frac{1}{2} (\sin (m + n) x + \sin (m - n) x) \\ = \frac{1}{2} (\sin (m x) \cos (n x) + \cos (m x) \sin (n x) + \sin (m x) \cos (n x) - \cos (m x) \sin (n x)) \\ = \sin (m x) \cos (n x) \end{array}$

$\begin{array}{l} \frac{1}{2} (\cos (m - n) x + \cos (m + n) x) \\ = \frac{1}{2} (\cos (m x) \cos (n x) - \sin (m x) \sin (n x) + \cos (m x) \cos (n x) - \sin (n x) \sin (n x)) \\ = \cos (m x) \cos (n x) \end{array}$

コード(Emacs)

Python 3

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

x, m, n = symbols('x, m, n')
eqs = [
    (cos((m - n) * x) - cos((m + n) * x)) / 2,
    (sin((m + n) * x) + sin((m - n) * x)) / 2,
    (cos((m - n) * x) + cos((m + n) * x)) / 2,
]

for eq in eqs:
    for t in [eq.factor(), eq.simplify()]:
        pprint(t)
        print()
    print()

d = {m: 1, n: 2}
eqs = [eq.subs(d) for eq in eqs]

p = plot(*eqs, (x, -5, 5), ylim=(-5, 5), legend=True, show=False)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample7.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample7.py
-(-cos(m⋅x - n⋅x) + cos(m⋅x + n⋅x)) 
────────────────────────────────────
                 2                  

sin(m⋅x)⋅sin(n⋅x)


sin(m⋅x - n⋅x) + sin(m⋅x + n⋅x)
───────────────────────────────
               2               

sin(m⋅x)⋅cos(n⋅x)


cos(m⋅x - n⋅x) + cos(m⋅x + n⋅x)
───────────────────────────────
               2               

cos(m⋅x)⋅cos(n⋅x)


$

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">
<br>
<label for="m0">m = </label>
<input id="m0" type="number" value="1">
<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="sample7.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 = [],
        fns = [[(x) => (Math.cos((m0 - n0) * x) - Math.cos((m0 + n0) * x)) / 2,
                'red'],
               [(x) => Math.sin(m0 * x) * Math.sin(n0 * x), 'green'],
               [(x) => (Math.sin((m0 - n0) * x) + Math.sin((m0 + n0) * x)) / 2,
                'blue'],
               [(x) => Math.sin(m0 * x) * Math.cos(n0 * x), 'orange'],
               [(x) => (Math.cos((m0 - n0) * x) + Math.cos((m0 + n0) * x)) / 2,
                'purple'],
               [(x) => Math.cos(m0 * x) * Math.cos(n0 * x), 'brown']],
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), 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, fns1, fns2].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 =
m = n =

Kamimura's blog

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2018年6月1日金曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(正弦、余弦、加法定理)

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