数学 - Python - JavaScript - 線型代数 - 複素数、複素ベクトル空間 - 二項方程式(複素数のn乗根、円の周、等分、極形式)

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2.を取り組んでみる。

2. 
   

   複素数の極形式を、

   $\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}=r\left(\cos \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+i\sin \mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\right)$

   とおく。

   この n 乗を根は、

   $\sqrt[n]{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}=\sqrt[n]{r}\left(\cos \frac{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}+i\sin \frac{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}\right)\left(k=0,1,\dots \; <!--\; horizontal\; ellipsis\; -->,n-1\right)$

   各根の偏角の差について考える。

   $\begin{array}{}\frac{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2\left(k+1\right)\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}-\frac{\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}+2k\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}=\frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}\\ \frac{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{n}·\; <!--\; middle\; dot\; -->n=2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\end{array}$

   よって、 問題の複素数の n 乗根は、 原点を中心とする半径

   $\sqrt[n]{r}$

   の周を n 等分する。

コード(Emacs)

Python 3

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

a, x = symbols('a, x', real=True)
n = symbols('n', integer=True)
z = a * (cos(x) + I * sin(x))
z0 = z ** (1 / n)

for t in [z0, z0.expand()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample2.py
n _______________________
╲╱ a⋅(ⅈ⋅sin(x) + cos(x)) 

n _______________________
╲╱ ⅈ⋅a⋅sin(x) + a⋅cos(x) 

$

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="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="r1">r1 = </label>
<input id="r1" type="number" min="0" value="1000000">
<label for="θ1">θ1 = </label>
<input id="θ1" type="number" min="0" value="1.5">
<label for="n1">n1 = </label>
<input id="n1" type="number" min="0" step="1" value="10">

<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="sample2.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_r1 = document.querySelector('#r1'),
    input_θ1 = document.querySelector('#θ1'),
    input_n1 = document.querySelector('#n1'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_r1, input_θ1, input_n1],
    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 = (n, r, θ) => (r ** (1 / n)) * Math.cos(θ),
    fy = (n, r, θ) => (r ** (1 / n)) * Math.sin(θ);

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),
        r1 = parseFloat(input_r1.value),
        θ1 = parseFloat(input_θ1.value),
        n1 = parseInt(input_n1.value, 10);

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

    let points = [],
        lines =
        range(0, n1)
        .map((k) => [0, 0,
                     fx(n1, r1, θ1 + 2 * k * Math.PI / n1),
                     fy(n1, r1, θ1 + 2 * k * Math.PI / n1),
                     'green']),
        g1 = (x) => Math.sqrt((r1 ** (1 / n1)) ** 2 - x ** 2),
        g2 = (x) => -g1(x),
        fns = [[g1, 'red'],
               [g2, '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 =
r1 = θ1 = n1 =