数学 - Python - 基本理論(Basic Theory) - 積分(Integration) - 冪関数の積分

学習環境

Surface 3 (4G LTE)、Surface 3 タイプカバー、Surface ペン(端末)
Windows 10 Pro (OS)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

オイラーの贈物―人類の至宝eiπ=-1を学ぶ (吉田武(著)、東海大学出版会)の第Ⅰ部(基礎理論(Basic Theory))、第4章(積分(Integration))、4.3(冪関数の積分)、問題1.を取り組んでみる。

1. $\begin{array}{l} 1 - \frac{1}{2!} x^{2} = 0 \\ x = \pm \sqrt{2} \\ \int_{- \sqrt{2}}^{\sqrt{2}} (1 - \frac{1}{2!} x^{2}) d x \\ = 2 \int_{0}^{\sqrt{2}} (1 - \frac{1}{2!} x^{2}) d x \\ = 2 {[x - \frac{1}{6} x^{3}]}_{0}^{\sqrt{2}} \\ = 2 (\sqrt{2} - \frac{2 \sqrt{2}}{6}) \\ = \frac{4 \sqrt{2}}{3} \end{array}$
2. $\begin{array}{l} 1 - \frac{1}{2!} x^{2} + \frac{1}{4!} x^{4} = 0 \\ x^{4} - 12 x^{2} + 24 = 0 \\ x^{2} = 6 \pm \sqrt{36 - 24} = 6 \pm 2 \sqrt{3} \\ x = \pm \sqrt{6 \pm 2 \sqrt{3}} \\ \int_{- \sqrt{6 - 2 \sqrt{3}}}^{\sqrt{6 - 2 \sqrt{3}}} (1 - \frac{1}{2!} x^{2} + \frac{1}{4!} x^{4}) d x \\ = 2 \int_{0}^{\sqrt{6 - 2 \sqrt{3}}} (1 - \frac{1}{2!} x^{2} + \frac{1}{4!} x^{4}) d x \\ = 2 {[x - \frac{1}{3!} x^{3} + \frac{1}{5!} x^{5}]}_{0}^{\sqrt{6 - 2 \sqrt{3}}} \\ = 2 \sqrt{6 - 2 \sqrt{3}} (1 - \frac{6 - 2 \sqrt{3}}{6} + \frac{{(6 - 2 \sqrt{3})}^{2}}{120}) \\ = 2 \sqrt{6 - 2 \sqrt{3}} (1 - \frac{3 - \sqrt{3}}{3} + \frac{4 {(3 - \sqrt{3})}^{2}}{120}) \\ = 2 \sqrt{6 - 2 \sqrt{3}} (1 - \frac{3 - \sqrt{3}}{3} + \frac{{(3 - \sqrt{3})}^{2}}{30}) \\ = 2 \sqrt{6 - 2 \sqrt{3}} (\frac{30 - 30 + 10 \sqrt{3} + 9 + 3 - 6 \sqrt{3}}{30}) \\ = \sqrt{6 - 2 \sqrt{3}} \frac{4 \sqrt{3} + 12}{15} \\ = \frac{4 \sqrt{2 (3 - \sqrt{3})} (\sqrt{3} + 3)}{15} \\ = \frac{4 \sqrt{2 (3 - \sqrt{3}) (3 + 9 + 6 \sqrt{3})}}{15} \\ = \frac{4 \sqrt{2 (3 - \sqrt{3}) 6 (2 + \sqrt{3})}}{15} \\ = \frac{8 \sqrt{3 (6 - 3 + \sqrt{3})}}{15} \\ = \frac{8 \sqrt{3 (3 + \sqrt{3})}}{15} \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, factorial, Rational, sqrt, solve, plot

print('1.')
x = symbols('x')
fs = [(1 - Rational(1, factorial(2)) * x ** 2, (-sqrt(2), sqrt(2))),
      (1 - Rational(1, factorial(2)) * x ** 2 + Rational(1, factorial(4)) * x ** 4, (-sqrt(6 - 2 * sqrt(3)), sqrt(6 - 2 * sqrt(3))))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'[{i}]')
    pprint(solve(f, dict=True))
    I = Integral(f, (x, x1, x2))
    for j, g in enumerate([I, I.doit()]):
        pprint(g.factor())

    p = plot(f, show=False, legend=True)
    p.save(f'sample1_{i}.svg')
    print()

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
[1]
[{x: -√2}, {x: √2}]
  √2             
  ⌠              
  ⎮  ⎛ 2    ⎞    
- ⎮  ⎝x  - 2⎠ dx 
  ⌡              
 -√2             
─────────────────
        2        
4⋅√2
────
 3  

[2]
⎡⎧      ___________⎫  ⎧     ___________⎫  ⎧      __________⎫  ⎧     __________
⎢⎨x: -╲╱ -2⋅√3 + 6 ⎬, ⎨x: ╲╱ -2⋅√3 + 6 ⎬, ⎨x: -╲╱ 2⋅√3 + 6 ⎬, ⎨x: ╲╱ 2⋅√3 + 6 
⎣⎩                 ⎭  ⎩                ⎭  ⎩                ⎭  ⎩               

⎫⎤
⎬⎥
⎭⎦
  ___________                      
╲╱ -2⋅√3 + 6                       
      ⌠                            
      ⎮        ⎛ 4       2     ⎞   
      ⎮        ⎝x  - 12⋅x  + 24⎠ dx
      ⌡                            
   ___________                     
-╲╱ -2⋅√3 + 6                      
───────────────────────────────────
                 24                
       _________         
4⋅√2⋅╲╱ -√3 + 3 ⋅(√3 + 3)
─────────────────────────
            15           

$

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="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<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="sample1.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let factorial = (n) => range(1, n + 1).reduce((prev, next) => prev * next, 1),
    f2 = (x) => 1 - 1 / factorial(2) * x ** 2 + 1 / factorial(4) * x ** 4;

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 = [],
        x3 = Math.sqrt(6 - 2 * Math.sqrt(3)),
        lines = [[-x3, y1, -x3, y2, 'red'],
                 [x3, y1, x3, y2, 'red']],
        fns = [[f2, 'green']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });                 

    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), 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, fns1, fns2].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

2017年8月23日水曜日

数学 - Python - 基本理論(Basic Theory) - 積分(Integration) - 冪関数の積分

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