## 2018年10月24日水曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - スターリングの公式(定理の証明4-6(不等式の証明))

1. 定理の証明の3より、

$\begin{array}{}\left(\frac{1}{2}\mathrm{log}\frac{1+x}{1-x}-x\right)-\frac{{x}^{3}}{3\left(1-{x}^{2}\right)}=\psi \left(x\right)\\ \psi \left(0\right)=0\\ \frac{1}{2}\mathrm{log}\frac{1+x}{1-x}-x=\frac{{x}^{3}}{\left(1-{x}^{2}\right)}=0\left(x=0\right)\\ 0

よって、 x が 0以上1未満の場合、

$0\le \frac{1}{2}\mathrm{log}\frac{1+x}{1-x}-x\le \frac{{x}^{3}}{3\left(1-{x}^{2}\right)}$

が成り立つ。

2. $\begin{array}{}\frac{1+x}{1-x}\\ =\frac{1+\frac{1}{2n+1}}{1-\frac{1}{2n+1}}\\ =\frac{2n+1+1}{2n+1-1}\\ =\frac{2n+2}{2n}\\ =\frac{n+1}{n}\end{array}$

また、

$\begin{array}{}\frac{{x}^{3}}{3\left(1-{x}^{2}\right)}\\ =\frac{{\left(\frac{1}{2n+1}\right)}^{3}}{3·\left(1-{\left(\frac{1}{2n+1}\right)}^{2}\right)}\\ =\frac{1}{3\left(2n+1\right)\left({\left(2n+1\right)}^{2}-1\right)}\\ =\frac{1}{3\left(2n+1\right)\left(4{n}^{2}+4n\right)}\\ =\frac{1}{12\left(2n+1\right)\left({n}^{2}+n\right)}\end{array}$

3. 4、5 より、

$0\le \frac{1}{2}\mathrm{log}\frac{n+1}{n}-\frac{1}{2n+1}\le \frac{1}{12\left(2n+1\right)\left({n}^{2}+n\right)}$

よって、

$\begin{array}{}0·\left(2n+1\right)\le \left(2n+1\right)\left(\frac{1}{2}\mathrm{log}\frac{n+1}{n}-\frac{1}{2n+1}\right)\le \left(2n+1\right)·\frac{1}{12\left(2n+1\right)\left({n}^{2}+n\right)}\\ 0\le \left(n+\frac{1}{2}\right)\mathrm{log}\frac{n+1}{n}-1\le \frac{1}{12\left({n}^{2}+n\right)}=\frac{1}{12}\left(\frac{1}{n\left(n+1\right)}\right)=\frac{1}{12}\left(\frac{1}{n}-\frac{1}{n+1}\right)\end{array}$

が成り立つ。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, log, Derivative, plot

print('4.')

x = symbols('x')
f = Rational(1, 2) * log((1 + x) / (1 - x)) - x

g = x ** 3 / (3 * (1 - x ** 2))

p = plot(f, g, (x, 0, 1), ylim=(-0.1, 0.1), show=False, legend=True)
colors = ['red', 'green']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample2.svg')
print('5.')
n = symbols('n')
d = {x: 1 / (2 * n + 1)}
pprint(g.subs(d).factor())

print('6.')
for func in [f, g]:
fn = func.subs(d)
for t in [fn, fn.factor(), fn.expand(), fn.simplify()]:
pprint(t)
print()
print()


$./sample2.py 4. 5. 1 ────────────────────── 12⋅n⋅(n + 1)⋅(2⋅n + 1) 6. ⎛ 1 ⎞ ⎜1 + ───────⎟ ⎜ 2⋅n + 1⎟ log⎜───────────⎟ ⎜ 1 ⎟ ⎜1 - ───────⎟ ⎝ 2⋅n + 1⎠ 1 ──────────────── - ─────── 2 2⋅n + 1 ⎛ 1 1 ⎞ ⎛ 1 2⋅n⋅log⎜─────────────────────────── + ───────────⎟ + log⎜───────────────────── ⎜ 2⋅n 1 1 ⎟ ⎜ 2⋅n ⎜2⋅n - ─────── + 1 - ─────── 1 - ───────⎟ ⎜2⋅n - ─────── + 1 - ─ ⎝ 2⋅n + 1 2⋅n + 1 2⋅n + 1⎠ ⎝ 2⋅n + 1 2 ────────────────────────────────────────────────────────────────────────────── 2⋅(2⋅n + 1) 1 ⎞ ────── + ───────────⎟ - 2 1 1 ⎟ ────── 1 - ───────⎟ ⋅n + 1 2⋅n + 1⎠ ───────────────────────── ⎛ 1 1 ⎞ log⎜─────────────────────────── + ───────────⎟ ⎜ 2⋅n 1 1 ⎟ ⎜2⋅n - ─────── + 1 - ─────── 1 - ───────⎟ ⎝ 2⋅n + 1 2⋅n + 1 2⋅n + 1⎠ 1 ────────────────────────────────────────────── - ─────── 2 2⋅n + 1 ⎛n + 1⎞ (2⋅n + 1)⋅log⎜─────⎟ - 2 ⎝ n ⎠ ──────────────────────── 2⋅(2⋅n + 1) 1 ─────────────────────────── ⎛ 3 ⎞ 3 ⎜3 - ──────────⎟⋅(2⋅n + 1) ⎜ 2⎟ ⎝ (2⋅n + 1) ⎠ 1 ────────────────────── 12⋅n⋅(n + 1)⋅(2⋅n + 1) 1 ────────────────────────────────────────────────────────────────────────────── 3 2 3 24⋅n 2 36⋅n 18⋅n 24⋅n - ────────────── + 36⋅n - ────────────── + 18⋅n - ────────────── + 3 - 2 2 2 4⋅n + 4⋅n + 1 4⋅n + 4⋅n + 1 4⋅n + 4⋅n + 1 ────────────── 3 ────────────── 2 4⋅n + 4⋅n + 1 1 ───────────────────── ⎛ 2 ⎞ 12⋅n⋅⎝2⋅n + 3⋅n + 1⎠$


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="-2">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="2">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-2">
<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="sample2.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) => 1 / 2 * Math.log((1 + x) / (1 - x)) - x,
fns = [[f, 'red'],
[(x) => x ** 3 / (3 * (1 - x ** 2)), 'green']];

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 = [[0, y1, 0, y2, 'blue'],
[1, y1, 1, y2, 'brown']];

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();