2018年5月21日月曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 部分積分法(広義積分、累乗(べき乗、平方)、指数関数、逆数、面積、極限)

1. $\begin{array}{}{\int }_{1}^{t}{x}^{2}{e}^{\left(-x\right)}\mathrm{dx}\\ ={\left[-{x}^{2}{e}^{-x}\right]}_{1}^{t}+2{\int }_{1}^{t}x{e}^{\left(-x\right)}\mathrm{dx}\\ =-{t}^{2}{e}^{\left(-t\right)}+{e}^{\left(-1\right)}+2\left({\left[-x{e}^{\left(-x\right)}\right]}_{1}^{t}+{\int }_{1}^{t}{e}^{\left(-x\right)}\mathrm{dx}\right)\\ =-{t}^{2}{e}^{\left(-t\right)}+{e}^{\left(-1\right)}+2\left(-t{e}^{\left(-t\right)}+{e}^{\left(-1\right)}+{\left[-{e}^{\left(-x\right)}\right]}_{1}^{t}\right)\\ =-{t}^{2}{e}^{\left(-t\right)}+{e}^{\left(-1\right)}-2t{e}^{\left(-t\right)}+2{e}^{\left(-1\right)}+2\left(-{e}^{\left(-t\right)}+{e}^{\left(-1\right)}\right)\\ ={e}^{\left(-t\right)}\left(-{t}^{2}-2t-2\right)+5{e}^{\left(-1\right)}\end{array}$

よって広義積分は収束する。

$\begin{array}{}{\int }_{1}^{\infty }{x}^{2}{e}^{\left(-x\right)}\mathrm{dx}\\ =\underset{t\to \infty }{\mathrm{lim}}{\int }_{1}^{t}{x}^{2}{e}^{\left(-x\right)}\mathrm{dx}\\ =5{e}^{\left(-1\right)}\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
f = x**2 * exp(-x)
I = Integral(f, (x, 1, oo))
for t in [I, I.doit()]:
pprint(t)
print()

p = plot(x ** 2, exp(-x), f, ylim=(-10, 10), legend=True, show=False)

for i, color in enumerate(['red', 'green', 'blue']):
p[i].line_color = color

p.save('sample22.svg')


$./sample22.py ∞ ⌠ ⎮ 2 -x ⎮ x ⋅ℯ dx ⌡ 1 -1 5⋅ℯ$


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="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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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 fns = [[(x) => x ** 2, 'red'],
[(x) => Math.exp(-x), 'green'],
[(x) => x ** 2 * Math.exp(-x), 'blue']];

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 = [],
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();