## 2018年10月30日火曜日

### 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(定理の系の証明(階乗、累乗、指数関数、円周率、極限)

1. $\begin{array}{}\frac{\pi }{2}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{2}{1}·\frac{2}{3}·\frac{4}{3}·\frac{4}{5}·\frac{6}{5}·\frac{6}{7}·\dots ·\frac{2n}{2n-1}·\frac{2n}{2n+1}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{2}{4}^{2}·\dots ·{\left(2n-2\right)}^{2}}{{3}^{2}{5}^{2}·\dots ·{\left(2n-1\right)}^{2}}·\frac{{\left(2n\right)}^{2}}{2n+1}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{2}{4}^{2}·\dots ·{\left(2n-2\right)}^{2}}{{3}^{2}{5}^{2}·\dots ·{\left(2n-1\right)}^{2}}·\frac{2n}{1+\frac{1}{2n}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{2}{4}^{2}·\dots ·{\left(2n-2\right)}^{2}}{{3}^{2}{5}^{2}·\dots ·{\left(2n-1\right)}^{2}}·2n\underset{n\to \infty }{\mathrm{lim}}\frac{1}{1+\frac{1}{2n}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{2}{4}_{·}^{2}·\dots ·{\left(2n-2\right)}^{2}}{{3}^{2}{5}^{2}·\dots ·{\left(2n-1\right)}^{2}}·2n\end{array}$

平方根をとる。

$\begin{array}{}{\left(\frac{\pi }{2}\right)}^{\frac{1}{2}}=\underset{n\to \infty }{\mathrm{lim}}\frac{2·4·\dots ·\left(2n-2\right)}{3·5·\dots ·\left(2n-1\right)}·{\left(2n\right)}^{\frac{1}{2}}\\ {\pi }^{\frac{1}{2}}=\underset{n\to \infty }{\mathrm{lim}}\frac{2·4·6·\dots ·2n}{2·4·6·\dots ·2n}·\frac{2·4·\dots ·\left(2n-2\right)}{3·5·\dots ·\left(2n-1\right)}2{n}^{\frac{1}{2}}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{2}^{n}n!}{\left(2n\right)!}·{2}^{n-1}\left(n-1\right)!·2·n·{n}^{\left(-\frac{1}{2}\right)}\\ =\underset{n\to \infty }{\mathrm{lim}}\frac{{\left(n!\right)}^{2}{2}^{2n}}{\left(2n\right)!{n}^{\frac{1}{2}}}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, pi, plot, product, oo

n, k = symbols('n, k', integer=True)
f = product((2 * k / (2 * k + 1)) ** 2, (k, 1, n - 1)) * 2 * n
for t in [f, f.limit(n, oo)]:
pprint(t)

p = plot(pi / 2, f, (n, 1, 100), show=False, legend=True)
colors = ['red', 'green']
for i, color in enumerate(colors):
p[i].line_color = color

p.save('sample4.svg')


$./sample4.py 2 2⋅n⋅(n - 1)! ──────────────────────────── 2 RisingFactorial (3/2, n - 1) π ─ 2$


$./sample3.py 3. RisingFactorial(1/2, n)⋅RisingFactorial(3/2, n) ─────────────────────────────────────────────── 2 n! 2 ─ π$


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="1.5">
<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="sample4.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 => {
let n = Math.floor(x),
res = 2 * n;

for (let i = 1; i < n; i += 1) {
res *= (2 * i) ** 2 / (2 * i + 1) ** 2;
}

return res;
},
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, Math.PI / 2, x2, Math.PI / 2, '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();