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))

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)、3(ウォリスの公式)の定理3の証明3.を取り組んでみる。

$\begin{array}{l} \frac{\int_{0}^{\frac{π}{2}} \sin^{2 n} x dx}{\int_{0}^{\frac{π}{2}} \sin^{2 n + 1} x dx} \\ = \frac{2 n - 1}{2 n} \cdot \frac{2 n - 3}{2 n - 2} \cdot \dots \cdot \frac{1}{2} \cdot \frac{π}{2} \cdot \frac{2 n + 1}{2 n} \cdot \frac{2 n - 1}{2 n - 2} \cdot \dots \cdot \frac{3}{2} \\ = \frac{π}{2} \cdot \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{5}{6} \cdot \frac{7}{6} \cdot \dots \cdot \frac{2 n - 3}{2 n - 2} \cdot \frac{2 n - 1}{2 n - 2} \cdot \frac{2 n - 1}{2 n} \cdot \frac{2 n + 1}{2 n} \end{array}$

また、証明2より、

$1 \leq (\frac{\int_{0}^{\frac{π}{2}} \sin^{2 n} x dx}{\int_{0}^{\frac{π}{2}} \sin^{2 n + 1} x dx}) \leq (\frac{\int_{0}^{\frac{π}{2}} \sin^{2 n - 1} x dx}{\int_{0}^{\frac{π}{2}} \sin^{2 n + 1} x dx}) = 1 + \frac{1}{2 n}$

よって、

$\begin{array}{l} 1 \leq \lim_{n \to \infty} \frac{π}{2} \cdot \frac{1}{2} \frac{3}{2} . \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{5}{6} \cdot \frac{7}{6} \cdot \dots, \frac{2 n - 1}{2 n} \cdot \frac{2 n + 1}{2 n} \leq \lim (1 + \frac{1}{2 n}) = 1 \\ \lim_{n \to \infty} \frac{π}{2} \cdot \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{3}{4} \cdot \frac{5}{4} - \frac{5}{6} \cdot \frac{7}{6} \cdot \dots \cdot \frac{2 n - 1}{2 n} \cdot \frac{2 n + 1}{2 n} = 1 \\ \frac{2}{π} = \lim_{n \to \infty} \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{3}{4} \cdot \frac{5}{4} - \frac{5}{6} - \frac{7}{6} \cdot \dots \cdot \frac{2 n - 1}{2 n} \cdot \frac{2 n + 1}{2 n} \end{array}$

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')

入出力結果(Terminal, Jupyter(IPython))

$ ./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,
    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 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])
        .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();

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

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))

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