数学 - Python - JavaScript - オイラーの公式とその応用(Euler's Formula & Its Applications) - オイラーの公式(Euler's Formula) - オイラーの公式の応用 - 指数法則の利用: 加法定理の導出(三角関数(正弦、余弦)、無限級数の和、実部と虚部)

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III部(オイラーの公式とその応用(Euler's Formula & Its Applications))、第8章(オイラーの公式(Euler's Formula))、8.2(オイラーの公式の応用)、8.2.3(指数法則の利用: 加法定理の導出)、問題4.を取り組んでみる。

$\begin{array}{l} A + i B \\ = \sum_{k = 0}^{\infty} a^{k} \cos k θ + i \sum_{k = 0}^{\infty} a^{k} \sin k θ \\ = \sum_{k = 0}^{\infty} a^{k} e^{i k θ} \\ = \sum_{k = 0}^{\infty} {(a e^{i θ})}^{k} \\ = \lim_{n \to \infty} \sum_{k = 0}^{n} {(a e^{i θ})}^{k} \\ = \lim_{n \to \infty} \frac{1 - {(a e^{i θ})}^{n + 1}}{1 - a e^{i θ}} \\ = \frac{1}{1 - a e^{i θ}} \\ = \frac{1 - a e^{- i θ}}{(1 - a e^{i θ}) (1 - a e^{- i θ})} \\ = \frac{1 - a (\cos (- θ) + i \sin (- θ))}{1 + a^{2} - a (e^{i θ} + e^{- i θ})} \\ = \frac{1 - a (\cos θ - i \sin θ)}{1 + a^{2} - a (\cos θ + i \sin θ + \cos (- θ) + i \sin (- θ))} \\ = \frac{(1 - a \cos θ) + i a \sin θ}{1 + a^{2} - a (\cos θ + i \sin θ + \cos θ - i \sin θ)} \\ = \frac{(1 - a \cos θ) + i a \sin θ}{1 + a^{2} - 2 a \cos θ} \\ = \frac{1 - a \cos θ}{1 + a^{2} - 2 a \cos θ} + i \frac{a \sin θ}{1 + a^{2} - 2 a \cos θ} \end{array}$
よって、実部と虚部を比較して、次のことが成り立つ。
$\begin{array}{l} A = \frac{1 - a \cos θ}{1 + a^{2} - 2 a \cos θ} \\ B = \frac{a \sin θ}{1 + a^{2} - 2 a \cos θ} \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import symbols, pprint, sin, cos, oo, summation

print('問題4.')
a, θ = symbols('a θ')
k = symbols('k', integer=True)
A = summation(a ** k * cos(k * θ), (k, 0, oo))
B = summation(a ** k * sin(k * θ), (k, 0, oo))
for t in [A, B]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
問題4.
  ∞              
 ___             
 ╲               
  ╲    k         
  ╱   a ⋅cos(k⋅θ)
 ╱               
 ‾‾‾             
k = 0            

  ∞              
 ___             
 ╲               
  ╲    k         
  ╱   a ⋅sin(k⋅θ)
 ╱               
 ‾‾‾             
k = 0            

$

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="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="0">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<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">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" min="-1" max="1" value="0.8">

<label for="θ0">θ = </label>
<input id="θ0" type="number" min="-1" max="1" 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="sample4.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_a0 = document.querySelector('#a0'),
    input_θ0 = document.querySelector('#θ0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_θ0],
    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 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),
        a0 = parseFloat(input_a0.value),
        θ0 = parseFloat(input_θ0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        A = (1 - a0 * Math.cos(θ0)) / (1 + a0 ** 2 - 2 * a0 * Math.cos(θ0)),
        lines = [[x1, A, x2, A, 'red']],
        term = (k) => a0 ** k * Math.cos(k * θ0),
        f = (x) =>
        range(0, Math.floor(x) + 1)
        .map((k) => term(k))
        .reduce((x, y) => x + y, 0),
        fns = [[f, '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 =
a = θ =

Kamimura's blog

2017年10月22日日曜日

数学 - Python - JavaScript - オイラーの公式とその応用(Euler's Formula & Its Applications) - オイラーの公式(Euler's Formula) - オイラーの公式の応用 - 指数法則の利用: 加法定理の導出(三角関数(正弦、余弦)、無限級数の和、実部と虚部)

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