## 2018年10月18日木曜日

### 数学 - Python - JavaScript - 解析学 - 積分 - いくつかの計算練習 - (n!)^(1/n)の計算(階乗、累乗、指数関数、不等号、対数関数の定積分、上方和および下方和との比較)

1. $\begin{array}{}{\int }_{1}^{n}\mathrm{log}x\mathrm{dx}\\ ={\left[x\mathrm{log}x\right]}_{1}^{n}-\underset{1}{\overset{n}{\int }}1\mathrm{dx}\\ =n\mathrm{log}n-{\left[x\right]}_{1}^{n}\\ =n\mathrm{log}n-n+1\end{array}$

上方和。

$\sum _{k=2}^{n}\mathrm{log}k$

下方和。

$\sum _{k=1}^{n-1}\mathrm{log}k$

比較。

$\sum _{k=1}^{n-1}\mathrm{log}k\le n\mathrm{log}n-n+1\le \sum _{k=2}^{n}\mathrm{log}k$

e の肩にのせる。（指数関数）

$\begin{array}{}{e}^{\sum _{k=1}^{n-1}\mathrm{log}k}\le {e}^{\left(n\mathrm{log}n-n+1\right)}\le {e}^{\sum _{k=2}^{n}\mathrm{log}k}\\ \prod _{k=1}^{n-1}{e}^{\mathrm{log}k}\le {n}^{n}{e}^{-n}e\le \prod _{k=2}^{n}{e}^{\mathrm{log}k}\\ \left(n-1\right)!\le {n}^{n}{e}^{-n}e\le n!\end{array}$

コード(Emacs)

Python 3

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

print('定理1の証明.')

x = symbols('x', real=True)
n = symbols('n', integer=True, nonnegative=True)
I = Integral(log(x), (x, 1, n))

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

l = factorial(n - 1)
c = n ** n * exp(-n) * E
r = factorial(n)

for m in range(1, 10):
print(f'n = {m}')
d = {n:m}
lm = l.subs(d)
cm = c.subs(d)
rm = r.subs(d)
for t in [lm, cm, rm, lm <= cm <= rm]:
pprint(t)
print()

p = plot(l, c, r, (n, 1, 5), legend=True, show=False)
colors = ['red', 'green', 'blue']
for i, color in enumerate(colors):
p[i].line_color = color
p.save('sample0.svg')


$./sample0.py 定理1の証明. n ⌠ ⎮ log(x) dx ⌡ 1 n⋅log(n) - n + 1 n = 1 1 1 1 True n = 2 1 -1 4⋅ℯ 2 True n = 3 2 -2 27⋅ℯ 6 True n = 4 6 -3 256⋅ℯ 24 True n = 5 24 -4 3125⋅ℯ 120 True n = 6 120 -5 46656⋅ℯ 720 True n = 7 720 -6 823543⋅ℯ 5040 True n = 8 5040 -7 16777216⋅ℯ 40320 True n = 9 40320 -8 387420489⋅ℯ 362880 True$


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="1">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="50">

<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="sample0.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 => n < 1 ? 1 : n * factorial(n - 1);
fns = [[x => factorial(Math.floor(x) - 1), 'red'],
[x => {
let n = Math.floor(x);

return (n ** n) * Math.exp(-n) * Math.E;
}, 'green'],
[x => factorial(Math.floor(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 = [[1, y1, 1, y2, 'orange']];

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();