## 2018年10月29日月曜日

### 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(定理の証明1(三角関数(正弦)、比率、極限、はさみうちの原理))

1. $\begin{array}{}\frac{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n}x\mathrm{dx}}{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n+1}x\mathrm{dx}}\\ =\frac{2n-1}{2n}·\frac{2n-3}{2n-2}·\dots ·\frac{1}{2}·\frac{\pi }{2}·\frac{2n+1}{2n}·\frac{2n-1}{2n-2}·\dots ·\frac{3}{2}\\ =\frac{\pi }{2}·\frac{1}{2}·\frac{3}{2}·\frac{3}{4}·\frac{5}{4}·\frac{5}{6}·\frac{7}{6}·\dots ·\frac{2n-3}{2n-2}·\frac{2n-1}{2n-2}·\frac{2n-1}{2n}·\frac{2n+1}{2n}\end{array}$

また、証明2より、

$1\le \left(\frac{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n}x\mathrm{dx}}{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n+1}x\mathrm{dx}}\right)\le \left(\frac{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n-1}x\mathrm{dx}}{\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\mathrm{sin}}^{2n+1}x\mathrm{dx}}\right)=1+\frac{1}{2n}$

よって、

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

コード(Emacs)

Python 3

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

print('3.')

n, k = symbols('n, k', integer=True)
f = product((2 * k - 1) / (2 * k) * (2 * k + 1) / (2 * k), (k, 1, n))
pprint(f)
pprint(f.limit(n, oo))
# p = plot(*[sin(x) ** n for n in range(5)], (x, -2, 2), show=False, legend=True)
# colors = ['red', 'green', 'blue', 'orange', 'brown']
# for i, color in enumerate(colors):
#     p[i].line_color = color

# p.save('sample2.svg')


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


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="0.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1">

<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="sample3.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 = 1;

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

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, 2 / Math.PI, x2, 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();