## 2018年10月11日木曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - いろいろな問題(部分積分法、帰納法、階乗、指数関数、定積分)

1. $\begin{array}{}\int {\left(1-{x}^{2}\right)}^{n}\mathrm{dx}\\ =x{\left(1-{x}^{2}\right)}^{n}-\int xn\left(-2x\right){\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-{x}^{2}\right)}^{n}-2n\int \left(-{x}^{2}\right){\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-{x}^{2}\right)}^{n}-2n\int \left(-1+1-{x}^{2}\right){\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ =x{\left(1-{x}^{2}\right)}^{n}+2n\int {\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}-2n\int {\left(1-{x}^{2}\right)}^{n}\mathrm{dx}\end{array}$

よって、

$\begin{array}{}\left(2n+1\right)\int {\left(1-{x}^{2}\right)}^{n}\mathrm{dx}=x{\left(1-{x}^{2}\right)}^{n}+2n\int {\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ \int {\left(1-{x}^{2}\right)}^{n}\mathrm{dx}=\frac{1}{2n+1}x{\left(1-{x}^{2}\right)}^{n}+\frac{2n}{2n+1}\int {\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\end{array}$

ゆえに、

$\begin{array}{}{\int }_{0}^{1}\left(1-{x}^{2}\right)\mathrm{dx}\\ =\frac{1}{2n+1}{\left[x{\left(1-{x}^{2}\right)}^{n}\right]}_{0}^{1}+\frac{2n}{2n+1}{\int }_{0}^{1}{\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ =\frac{2n}{2n+1}{\int }_{0}^{1}{\left(1-{x}^{2}\right)}^{n-1}\mathrm{dx}\\ =\frac{2n}{2n+1}·\frac{{2}^{2\left(n-1\right)}{\left(\left(n-1\right)!\right)}^{2}}{\left(2\left(n-1\right)+1\right)!}\\ =\frac{n}{2n+1}·\frac{{2}^{2n-1}{\left(\left(n-1\right)!\right)}^{2}}{\left(2n-1\right)!}\\ =\frac{n}{2n+1}·\frac{2n}{2n}·\frac{{2}^{2n-1}{\left(\left(n-1\right)!\right)}^{2}}{\left(2n-1\right)!}\\ =\frac{{2}^{2n}{\left(n!\right)}^{2}}{\left(2n+1\right)!}\end{array}$

以上から、 帰納法により、すべての負でない整数に対して成り立つ。

コード(Emacs)

Python 3

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

print('6.')

x = symbols('x')
n = symbols('n', integer=True, nonegative=True)
f = (1 - x ** 2) ** n

I = Integral(f, (x, 0, 1))
for t in [I, I.doit()]:
pprint(t)
print()

g = 2 ** (2 * n) * factorial(n) ** 2 / factorial(2 * n + 1)
p = plot(f.subs({n: 2}), g.subs({n: 2}), legend=True, show=False)
colors = ['red', 'green']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample6.svg')

pprint(g)

for n0 in range(10):
print(f'n = {n0}:', I.subs({n: n0}).doit() == g.subs({n: n0}))


$./sample6.py 6. 1 ⌠ ⎮ n ⎮ ⎛ 2 ⎞ ⎮ ⎝- x + 1⎠ dx ⌡ 0 ┌─ ⎛1/2, -n │ ⎞ ├─ ⎜ │ 1⎟ 2╵ 1 ⎝ 3/2 │ ⎠ 2⋅n 2 2 ⋅n! ────────── (2⋅n + 1)! n = 0: True n = 1: True n = 2: True n = 3: True n = 4: True n = 5: True n = 6: True n = 7: True n = 8: True n = 9: True$


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="-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">

<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="sample6.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],
p = (x) => pre0.textContent += x + '\n';

let f = (x) => (1 - x ** 2) ** 2,
fns = [[f, 'red']],
factorial = (n) => {
let result = 1;
for (let i = 1; i <= n; i += 1) {
result *= i;
}
return result;
}

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 = [],
n0 = 2,
yn = 2 ** (2 * n0) * factorial(n0) ** 2 / factorial(2 * n0 + 1),
lines = [[x1, yn, x2, yn, 'green'],
[0, y1, 0, y2, 'blue'],
[1, y1, 1, y2, 'orange']];

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();