## 2018年6月4日月曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(余弦、正弦、倍角、加法定理、積、零)

1. $\begin{array}{}{\int }_{-\pi }^{\pi }\mathrm{sin}\left(mx\right)\mathrm{cos}\left(nx\right)\mathrm{dx}\\ =\frac{1}{2}{\int }_{-\pi }^{\pi }\left(\mathrm{sin}\left(m+n\right)x+\mathrm{sin}\left(m-n\right)x\right)\mathrm{dx}\end{array}$

よって、

$m=n$

の とき、

$\begin{array}{}m\ne 0\\ \frac{1}{2}{\int }_{-\pi }^{\pi }\mathrm{sin}\left(2m\right)x\mathrm{dx}\\ =-\frac{1}{2}{\left[\frac{\mathrm{cos}\left(2m\right)x}{2m}\right]}_{-\pi }^{\pi }\\ =-\frac{1}{4m}\left(\mathrm{cos}\left(2m\pi \right)-\mathrm{cos}\left(2m\left(-\pi \right)\right)\right)\\ =0\\ m=0\\ \frac{1}{2}{\int }_{-\pi }^{\pi }\mathrm{sin}0\mathrm{dx}\\ =0\end{array}$

また、

$m\ne n$

のとき、

$\begin{array}{}m+n\ne 0\\ \frac{1}{2}{\int }_{-\pi }^{\pi }\left(\mathrm{sin}\left(m+n\right)x+\mathrm{sin}\left(m-n\right)x\right)\mathrm{dx}\\ =-\frac{1}{2}{\left[\frac{\mathrm{cos}\left(m+n\right)x}{m+n}+\frac{\mathrm{cos}\left(m-n\right)x}{m-n}\right]}_{-\pi }^{\pi }\\ =0\\ \frac{n}{}+n=0\\ \frac{1}{2}{\int }_{-\pi }^{\pi }\left(\mathrm{sin}0+\mathrm{sin}\left(m-n\right)x\right)\mathrm{dx}\\ =-\frac{1}{2}{\left[\frac{\mathrm{cos}\left(m-n\right)x}{m-n}\right]}_{-\pi }^{\pi }\\ =0\end{array}$

ゆえに、

$\begin{array}{}m,n\in \text{ℤ}\\ {\int }_{-\pi }^{\pi }\mathrm{sin}\left(mx\right)\mathrm{cos}\left(nx\right)\mathrm{dx}=0\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
m, n = symbols('m, n', integger=True)
f = sin(m * x) * cos(n * x)
I = Integral(f, (x, -pi, pi))
for t in [I, I.doit()]:
pprint(t)
print()

p = plot(sin(-2 * x), cos(3 * x), f.subs({m: -2, n: 3}), (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('sample11.svg')


$./sample11.py π ⌠ ⎮ sin(m⋅x)⋅cos(n⋅x) dx ⌡ -π 0$


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="-2">
<label for="n0">n = </label>
<input id="n0" type="number" 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="sample11.js"></script>


JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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 = [[-Math.PI, y1, -Math.PI, y2, 'orange'],
[Math.PI, y1, Math.PI, y2, 'brown']];
fns = [[(x) => Math.sin(m0 * x), 'red'],
[(x) => Math.cos(n0 * x), 'green'],
[(x) => Math.sin(m0 * x) * Math.cos(n0 * x), 'blue']],
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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();