数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 二項方程式(累乗根、極形式、複素平面、図示)

線型代数入門

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

線型代数入門(松坂 和夫(著)、岩波書店)の第4章(複素数、複素ベクトル空間)、4(二項方程式)、問題3.を取り組んでみる。

3. 1. 
      
      $\begin{array}{}\sqrt{1+i}=\sqrt{2}\left(\cos \frac{\frac{1}{\sqrt{2}}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}+i\sin \frac{\frac{1}{\sqrt{2}}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}\right)\left(k=0,1\right)\\ \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}=\frac{1}{2\sqrt{2}},\frac{1+2\sqrt{2}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}\end{array}$
      
   2. 
      
      $\begin{array}{}\sqrt[3]{i}=\cos \frac{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3}+i\sin \frac{\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3}\left(k=0,1,2\right)\\ \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}=\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{6},\frac{5\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{6},\frac{3\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{2}\end{array}$
      
   3. 
      
      $\begin{array}{}\sqrt[4]{-2+2\sqrt{3}}i=\sqrt[4]{2}(\cos \frac{\frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{4}+i\sin \frac{\frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{4})\left(k=0,1,2,3\right)\\ \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}=\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{6},\frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3},\frac{7\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{6},\frac{5\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{3}\end{array}$
      
   4. 
      
      $\begin{array}{}\sqrt[5]{-1}=\cos \frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5}+i\sin \frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5}\left(k=0,1,2,3,4\right)\\ \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}=\frac{\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5},\frac{3\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5},\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->},\frac{7\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5},\frac{9\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{5}\end{array}$
      

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, I, Rational

zs = [sqrt(1 + I),
      I ** Rational(1, 3),
      (-2 + 2 * sqrt(3) * I) ** Rational(1, 4),
      (-1) ** Rational(1, 5)]

for i, z in enumerate(zs):
    print(f'({chr(ord("a") + i)})')
    pprint(z.factor())
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
(a)
  _______
╲╱ 1 + ⅈ 

(b)
6 ____
╲╱ -1 

(c)
4 ___ 4 ___________
╲╱ 2 ⋅╲╱ -1 + √3⋅ⅈ 

(d)
5 ____
╲╱ -1 

$

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="-1.5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="1.5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-1.5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="1.5">

<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="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'),
    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 f = (x) => Math.sqrt(1 - x ** 2),
    g = (x) => -f(x);

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 =
        range(0, 5)
        .map((k) => [0, 0,
                     Math.cos((Math.PI + 2 * k * Math.PI) / 5),
                     Math.sin((Math.PI + 2 * k * Math.PI) / 5)]),
        fns = [[f, 'red'],
               [g, 'red']],
        fns1 = [],
        fns2 = [];

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

                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 =