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) - オイラーの公式の導出(計算)

オイラーの贈物
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開発環境

• macOS Mojave - Apple (OS)
• Emacs (Text Editor)
• Windows 10 Pro (OS)
• Visual Studio Code (Text Editor)
• Python 3.7 (プログラミング言語)

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))、8.1(オイラーの公式の導出)、問題1.を取り組んでみる。

1. 
   $\begin{array}{l}{e}^{i\frac{\pi }{6}}=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\\ =\frac{\sqrt{3}}{2}+\frac{1}{2}i\\ \\ e^{i\frac{\pi }{4}}=\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\\ =\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\\ \\ e^{i\frac{\pi }{3}}=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\\ =\frac{1}{2}+\frac{\sqrt{3}}{2}i\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import symbols, pprint, cos, sin, I, pi, plot

print('問題1.')
def f(x):
    return cos(x) + I * sin(x)

for x in [pi / 6, pi / 4, pi / 3]:
    pprint(f(x))
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
問題1.
√3   ⅈ
── + ─
2    2

√2   √2⋅ⅈ
── + ────
2     2  

1   √3⋅ⅈ
─ + ────
2    2  

$

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">
<br>
<label for="t1">t1 = </label>
<input id="t1" type="number" value="-5">
<label for="t2">t2 = </label>
<input id="t2" type="number" value="5">
<input id="dt" type="number" min="0" step="0.0001" value="0.001">

<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'),
    input_t1 = document.querySelector('#t1'),
    input_t2 = document.querySelector('#t2'),
    input_dt = document.querySelector('#dt'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_t1, input_t2, input_dt],
    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 fx = (t) => Math.cos(t),
    fy = (t) => Math.sin(t);

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),
        t1 = parseFloat(input_t1.value),
        t2 = parseFloat(input_t2.value),
        dt = parseFloat(input_dt.value);

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

    let points = [],
        lines = [],
        fns = [],
        fns1 = [],
        fns2 = [];

    for (let t = t1; t <= t2; t += dt) {
        points.push([fx(t), fy(t), 'green']);
    }
    
    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 =
t1 = t2 =