## 2018年10月31日水曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - ウォリスの公式(スターリングの公式における定数(極限))

1. $\begin{array}{}c=\underset{n\to \infty }{\mathrm{lim}}\frac{{n}^{n+\frac{1}{2}}{e}^{-n}}{n!}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{\left(2n\right)}^{2n+\frac{1}{2}}{e}^{-2n}}{\left(2n\right)!}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{2}\sqrt[n]{2}{n}^{\left(2n+\frac{1}{2}\right)}{e}^{-2n}}{\left(2n\right)!}·\frac{{n}^{\frac{1}{2}}}{{n}^{\frac{1}{2}}}·\frac{{\left(n!\right)}^{2}}{{\left(n!\right)}^{2}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{\left(n!\right)}^{2}2\sqrt[\left(27\right)]{2}}{\left(2n\right)!{n}^{\frac{1}{2}}}·\frac{{n}^{2\left(n+\frac{1}{2}\right)}·{e}^{-2n}}{{\left(n!\right)}^{2}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{\left(n!\right)}^{2}{2}^{\left(2n\right)}2}{\left(2n\right)!{n}^{\frac{1}{2}}}{\left(\frac{{2}^{n+\frac{1}{2}}{e}^{-n}}{n!}\right)}^{2}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{\left(n!\right)}^{2}{2}^{\left(2n\right)}\sqrt{2}}{\left(2n\right)!{n}^{\frac{1}{2}}}·{\left(\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{n+\frac{1}{2}}{e}^{-n}}{n!}\right)}^{2}\\ =\sqrt{2}\pi {c}^{2}\end{array}$

よって、

$c=\frac{1}{\sqrt{2\pi }}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Limit, exp, factorial, Rational, oo, plot, sqrt, pi

n = symbols('n', integer=True)
f = (2 * n) ** (2 * n + Rational(1, 2)) * exp(-2 * n) / factorial(2 * n)
c = Limit(f, n, oo)

for t in [c, c.doit()]:
pprint(t)
print()

p = plot(f, 1 / sqrt(2 * pi), (n, 1, 50), show=False, legend=True)
colors = ['red', 'green']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample5.svg')


$./sample5.py ⎛ 2⋅n + 1/2 -2⋅n⎞ ⎜(2⋅n) ⋅ℯ ⎟ lim ⎜────────────────────⎟ n─→∞⎝ (2⋅n)! ⎠ √2 ──── 2⋅√π$


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="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="20">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" 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="sample5.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 factorial = n => {
let res = 1;

for (let i = 1; i <= n ; i += 1) {
res *= i;
}
return res;
},
f = x => {
let n = Math.floor(x);

return factorial(2 * n) ** (2 * n + 1 / 2) * Math.exp(-2 * n) /
factorial(2 * n);
},
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, 1 / Math.sqrt(2 * Math.PI),
x2, 1 / Math.sqrt(2 * Math.PI), '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])
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();