2018年10月25日木曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第12章(いくつかの計算練習)、2(スターリングの公式)の定理2の証明7.を取り組んでみる。


  1. e 1 12 n > 1 , a n > 0

    よって

    a n a n e 1 12 n = b n

    また、

    a n + 1 a n = n + 1 n + 1 + 1 2 e - n + 1 n + 1 ! · n ! n n + 1 2 e - n = n + 1 n + 1 2 e - 1 n ! . n ! n n + 1 2 = n + 1 n n + 1 2 e - 1

    対数を考える。

    log n + 1 n n + 1 2 e - 1 = n + 1 2 log n + 1 n - 1

    6より、これは0以上なので、

    n + 1 n n + 1 2 e - 1 1 a n + 1 a n 1

    また、

    b n + 1 b n = a n + 1 a n · 1 e 12 n + 1 e 1 12 n = a n + 1 a n · e 1 12 1 n + 1 - 1 n = n + 1 n n + 1 2 e - 1 · e - 1 12 1 n - 1 n + 1

    対数を考える。

    n + 1 2 log n + 1 n - 1 - 1 12 1 n - 1 n + 1

    6より

    n + 1 2 log n + 1 n - 1 - 1 12 1 n - 1 n + 1 0

    よって、

    b n + 1 b n 1

コード(Emacs)

Python 3

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

print('7.')

n = symbols('n', integer=True)

a = n ** (n + Rational(1, 2)) * exp(-n) / factorial(n)
b = a * exp(1 / (12 * n))

for m in range(1, 10):
    print(a.subs({n: m + 1}) / a.subs({n: m}) >= 1,
          b.subs({n: m + 1}) / b.subs({n: m}) <= 1)

p = plot(a, b, (n, 1, 100), show=False, legend=True)
colors = ['red', 'green']
for i, color in enumerate(colors):
    p[i].line_color = color
p.save('sample3.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
7.
True True
True True
True True
True True
True True
True True
True True
True True
True 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="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="0.365">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="0.4">

<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,
    padding = 50,
    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 n ** (n + 1 / 2) * Math.exp(-n) / factorial(n);
    },
    g = x => {
        let n = Math.floor(x);

        return f(x) * Math.exp(1 / (12 * n));
    },
    fns = [[f, 'red'],
           [g, '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 = [];
    
    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])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    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();







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