## 2018年6月15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(連続関数、フーリエ係数、正弦、絶対値、加法定理、2倍角)

1. $\begin{array}{}{c}_{0}=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\left|\mathrm{sin}x\right|\mathrm{dx}\\ =\frac{1}{2\pi }·2·{\int }_{0}^{\pi }\mathrm{sin}x\mathrm{dx}\\ =\frac{1}{\pi }{\left[-\mathrm{cos}x\right]}_{0}^{\pi }\\ =\frac{1}{\pi }\left(-\mathrm{cos}\pi +\mathrm{cos}0\right)\\ =\frac{2}{\pi }\end{array}$
$\begin{array}{}{a}_{n}=\frac{1}{\pi }{\int }_{-\pi }^{\pi }\left|\mathrm{sin}x\right|\mathrm{cos}\left(nx\right)\mathrm{dx}\\ =\frac{1}{\pi }\left(-{\int }_{-\pi }^{0}\mathrm{sin}x\mathrm{cos}\left(nx\right)\mathrm{dx}+{\int }_{0}^{\pi }\mathrm{sin}x\mathrm{cos}\left(nx\right)\mathrm{dx}\right)\\ =\frac{1}{\pi }\left(-{\int }_{-\pi }^{0}\frac{\mathrm{sin}\left(1+n\right)x+\mathrm{sin}\left(1-n\right)x}{2}\mathrm{dx}+{\int }_{0}^{\pi }\frac{\mathrm{sin}\left(1+n\right)x+\mathrm{sin}\left(1-n\right)x}{2}\mathrm{dx}\right)\\ n=1\\ \frac{1}{2\pi }\left(-{\int }_{-\pi }^{0}\mathrm{sin}2x\mathrm{dx}+{\int }_{0}^{\pi }\mathrm{sin}2x\mathrm{dx}\right)\\ =0\\ n\ne 1\end{array}$
$\begin{array}{}\frac{1}{2\pi }\left({\left[\frac{\mathrm{cos}\left(1+n\right)x}{1+n}+\frac{\mathrm{cos}\left(1-n\right)x}{1-n}\right]}_{\left(-\pi \right)}^{0}-{\left[\frac{\mathrm{cos}\left(1+n\right)x}{1+n}+\frac{\mathrm{cos}\left(1-n\right)x}{1-n}\right]}_{0}^{\pi }\right)\\ =\frac{1}{2\pi }\left(\frac{2}{1+n}+\frac{2}{1-n}+\frac{2{\left(-1\right)}^{n}}{1+n}+\frac{2{\left(-1\right)}^{n}}{1-n}\right)\\ =\frac{2\left(1+{\left(-1\right)}^{n}\right)}{\pi \left(1-{n}^{2}\right)}\end{array}$
$\begin{array}{}{b}_{n}=\frac{1}{\pi }\left(-{\int }_{-\pi }^{0}\mathrm{sin}x\mathrm{sin}\left(nx\right)\mathrm{dx}+{\int }_{0}^{\pi }\mathrm{sin}x\mathrm{sin}\left(nx\right)\mathrm{dx}\right)\\ =\frac{1}{\pi }\left(-{\int }_{-\pi }^{0}\frac{\mathrm{cos}\left(1-n\right)x-\mathrm{cos}\left(1+n\right)x}{2}+{\int }_{0}^{\pi }\frac{\mathrm{cos}\left(1-n\right)x-\mathrm{cos}\left(1+n\right)x}{2}\mathrm{dx}\right)\\ n=1\\ \frac{1}{2\pi }\left(-{\left[x\right]}_{-\pi }^{0}+{\left[x\right]}_{0}^{\pi }\right)=0\\ n\ne 1\\ 0\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
n = symbols('n', positive=True, integer=True)

f = Function('f')(x)
c0 = 1 / (2 * pi) * (Integral(-f, (x, -pi, 0)) + Integral(f, (x, 0, pi)))
an = 1 / pi * (Integral(-f * cos(n * x), (x, -pi, 0)) +
Integral(f * cos(n * x), (x, 0, pi)))
bn = 1 / pi * (Integral(-f * sin(n * x), (x, -pi, 0)) +
Integral(f * sin(n * x), (x, 0, pi)))

fs = [c0, an, bn]

for t in fs:
g = t.subs({f: sin(x)})
for s in [g, g.doit(), g.doit().simplify()]:
pprint(s)
print()
print()

p = plot(abs(sin(x)), abs(sin(x)) * cos(2 * x),
abs(sin(x)) * sin(2 * x), legend=True, show=False)
for i, color in enumerate(['red', 'green', 'blue']):
p[i].line_color = color
p.save('sample22.svg')


$./sample22.py 0 π ⌠ ⌠ ⎮ -sin(x) dx + ⎮ sin(x) dx ⌡ ⌡ -π 0 ─────────────────────────── 2⋅π 2 ─ π 2 ─ π 0 π ⌠ ⌠ ⎮ -sin(x)⋅cos(n⋅x) dx + ⎮ sin(x)⋅cos(n⋅x) dx ⌡ ⌡ -π 0 ───────────────────────────────────────────── π ⎛⎧ 0 for n = 1⎞ ⎛⎧ 0 for n = 1⎞ ⎛⎧ 0 for n = 1⎞ ⎜⎪ ⎟ ⎜⎪ ⎟ ⎜⎪ ⎟ ⎜⎪ n ⎟ ⎜⎪ 1 ⎟ ⎜⎪ n ⎟ ⎜⎨-(-1) ⎟ - ⎜⎨────── otherwise⎟ + ⎜⎨ (-1) 1 ⎟ ⎜⎪─────── otherwise⎟ ⎜⎪ 2 ⎟ ⎜⎪- ────── - ────── otherwise⎟ ⎜⎪ 2 ⎟ ⎜⎪n - 1 ⎟ ⎜⎪ 2 2 ⎟ ⎝⎩ n - 1 ⎠ ⎝⎩ ⎠ ⎝⎩ n - 1 n - 1 ⎠ ────────────────────────────────────────────────────────────────────────────── π ⎧ 0 for n = 1 ⎪ ⎪ ⎛ n ⎞ ⎨-⎝2⋅(-1) + 2⎠ ⎪─────────────── otherwise ⎪ ⎛ 2 ⎞ ⎩ π⋅⎝n - 1⎠ 0 π ⌠ ⌠ ⎮ -sin(x)⋅sin(n⋅x) dx + ⎮ sin(x)⋅sin(n⋅x) dx ⌡ ⌡ -π 0 ───────────────────────────────────────────── π ⎛⎧-π ⎞ ⎛⎧π ⎞ ⎜⎪─── for n = 1⎟ ⎜⎪─ for n = 1⎟ ⎜⎨ 2 ⎟ + ⎜⎨2 ⎟ ⎜⎪ ⎟ ⎜⎪ ⎟ ⎝⎩ 0 otherwise⎠ ⎝⎩0 otherwise⎠ ─────────────────────────────────── π 0$


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="n0">n = </label>
<input id="n0" type="number" min="1" 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="sample22.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_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';

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 = 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.abs(Math.sin(x)), 'red'],
[(x) => Math.abs(Math.sin(x)) * Math.cos(n0 * x), 'green'],
[(x) => Math.abs(Math.sin(x)) * Math.sin(n0 * x), 'blue']];

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])
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].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();