## 2018年10月9日火曜日

### 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、広義積分、累乗関数、指数関数、積、帰納法)

1. $\begin{array}{}{I}_{0}=\underset{0}{\overset{\infty }{\int }}{x}^{0}{e}^{-x}\mathrm{dx}\\ =\underset{0}{\overset{\infty }{\int }}{e}^{-x}\mathrm{dx}\\ ={\left[-{e}^{-x}\right]}_{0}^{\infty }\\ =1\\ {I}_{1}=\underset{0}{\overset{\infty }{\int }}{x}^{1}{e}^{\left(-x\right)}\mathrm{dx}\\ ={\left[-x{e}^{\left(-x\right)}\right]}_{0}^{\infty }+\underset{0}{\overset{\infty }{\int }}{e}^{\left(-x\right)}\mathrm{dx}\\ =1\\ {I}_{2}=\underset{0}{\overset{\infty }{\int }}{x}^{2}{e}^{-x}\mathrm{dx}\\ ={\left[-{x}^{2}{e}^{-x}\right]}_{0}^{\infty }+2\underset{0}{\overset{\infty }{\int }}x{e}^{-x}\mathrm{dx}\\ =2\\ {I}_{3}=\underset{0}{\overset{\infty }{\int }}{x}^{3}{e}^{-x}\mathrm{dx}\\ ={\left[-{x}^{3}{e}^{-x}\right]}_{0}^{\infty }+3\underset{0}{\overset{\infty }{\int }}{x}^{2}{e}^{-x}\mathrm{dx}\\ =3·2\\ =6\end{array}$

以上から

${I}_{n}=n!$

と予想。

$\begin{array}{}{I}_{n}=\underset{0}{\overset{\infty }{\int }}{x}^{n}{e}^{-x}\mathrm{dx}\\ ={\left[-{x}^{n}{e}^{-x}\right]}_{0}^{\infty }+n\underset{0}{\overset{\infty }{\int }}{x}^{\left(n-1\right)}{e}^{\left(-x\right)}\mathrm{dx}\\ =n\left(n-1\right)!\end{array}$

ゆえに、帰納法より、推測通り、

$\underset{0}{\overset{\infty }{\int }}{x}^{n}{e}^{-x}\mathrm{dx}=n!$

コード(Emacs)

Python 3

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

print('4.')

x = symbols('x')
n = symbols('n', integer=True, nonnegative=True)
f = x ** n * exp(-x)
I = Integral(f, (x, 0, oo))
d = {n: 3}
for t in [I, I.doit()]:
pprint(t)
print()
p = plot(f.subs(d), I.doit().subs(d), xlim=(0, 10),
ylim=(0, 10), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
p[i].line_color = c
p.save('sample4.svg')


$./sample4.py 4. ∞ ⌠ ⎮ n -x ⎮ x ⋅ℯ dx ⌡ 0 Γ(n + 1)$


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="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample4.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_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) => x ** 3 * Math.exp(-x),
fns = [[f, 'red']];

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 = [],
lines = [[x1, 6, x2, 6, 'green']];

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();