数学 - Python - JavaScript - 解析学 - 各種の初等関数 - 複素数の幾何学的表現(複素平面、極形式、二次方程式、平面幾何学への応用)

解析入門〈1〉

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, Google Chrome...)用JavaScript Library: MathJax
• 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

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.5(複素数の幾何学的表現)、問題1、2、3、4、5、6.を取り組んでみる。

1. 
   $\begin{array}{l}x軸\\ \overline{\alpha }\\ y軸\\ -\overline{\alpha }\\ y=x\\ i\overline{\alpha }\\ y=-x\\ -i\overline{\alpha }\end{array}$
2. 
   $\begin{array}{l}\frac{\alpha -\gamma }{a\text{'}-\gamma \text{'}}=\frac{\alpha -\beta }{\alpha \text{'}-\beta \text{'}}\\ \frac{\alpha -\gamma }{\alpha -\beta }=\frac{\alpha \text{'}-\gamma \text{'}}{\alpha \text{'}-\beta \text{'}}\end{array}$
3. 
   $\begin{array}{l}\frac{\alpha -\gamma }{\alpha -\beta }=\frac{\beta -\alpha }{\beta -\gamma }\\ \alpha \beta -\beta \gamma -\alpha \gamma +{\gamma }^2=\alpha \beta -{\alpha }^2-{\beta }^2+\alpha \beta \\ {\alpha }^2+{\beta }^2+{\gamma }^2=\alpha \beta +\beta \gamma +\gamma \alpha \end{array}$
4. 
   $\begin{array}{l}\cos 4\theta +i\sin 4\theta ={\left(\cos \theta +i\sin \theta \right)}^4\\ ={\cos }^4\theta +4{\cos }^3\theta i\sin \theta -6{\cos }^2\theta \sin \theta -4i\cos \theta {\sin }^3\theta +{\sin }^4\theta \\ \cos 4\theta ={\cos }^4\theta -6{\cos }^2\theta \sin \theta +{\sin }^4\theta \\ \sin 4\theta =4{\cos }^3\theta \sin \theta -4\cos \theta {\sin }^3\theta \\ \\ \cos 5\theta +i\sin 5\theta ={\left(\cos \theta +i\sin \theta \right)}^5\\ ={\cos }^5\theta +5i{\cos }^4\theta \sin \theta -10{\cos }^3\theta {\sin }^2\theta -10i{\cos }^2\theta {\sin }^3\theta +5\cos \theta {\sin }^4\theta -i{\sin }^5\theta \\ \cos 5\theta ={\cos }^5\theta -10{\cos }^3\theta {\sin }^2\theta +5\cos \theta {\sin }^4\theta \\ \sin 5\theta =5{\cos }^4\theta \sin \theta -10{\cos }^2\theta {\sin }^3\theta -{\sin }^5\theta \end{array}$
5. 
   $\begin{array}{l}h=kn\\ n\\ h\ne kn\\ 0\end{array}$
6. 
   $\begin{array}{l}z=\cos \theta +i\sin \theta \\ \frac{1-z^{n+1}}{1-z}=\frac{1-\left(\cos \left(n+1\right)\theta +i\sin \left(n+1\right)\theta \right)}{1-\left(\cos \theta +i\sin \theta \right)}\\ =\frac{\left(1-\cos \left(n+1\right)\theta \right)-i\sin \left(n+1\right)\theta }{\left(1-\cos \theta \right)-i\sin \theta }\\ =\frac{\left(\left(1-\cos \left(n+1\right)\theta \right)-i\sin \left(n+1\right)\theta \right)\left(\left(1-\cos \theta \right)+i\sin \theta \right)}{{\left(1-\cos \theta \right)}^2+{\sin }^2\theta }\\ =\frac{\left(1-\cos \left(n+1\right)\theta \right)\left(1-\cos \theta \right)+\sin \left(n+1\right)\theta \sin \theta +i\left(\left(1-\cos \left(n+1\right)\theta \right)\sin \theta -\sin \left(n+1\right)\theta \left(1-\cos \theta \right)\right)}{1-2\cos \theta +{\cos }^2\theta +{\sin }^2\theta }\\ =\frac{1+\cos \left(n+1\right)\theta \cos \theta +\sin \left(n+1\right)\theta \sin \theta +i\left(\sin \theta -\cos \left(n+1\right)\theta \sin \theta -\sin \left(n+1\right)\theta +\sin \left(n+1\right)\theta \cos \theta \right)}{2\left(1-\cos \theta \right)}\\ =\frac{1+\cos \left(\left(n+1\right)-\theta \right)+i\left(\sin \theta -\sin \left(n+1\right)\theta +\sin \left(\left(n+1\right)-\theta \right)\right)}{2\left(1-\cos \theta \right)}\\ \frac{1+\cos \left(\left(n+1\right)-\theta \right)}{2\left(1-\cos \theta \right)}\\ \frac{\sin \theta -\sin \left(n+1\right)\theta +\sin \left(\left(n+1\right)-\theta \right)}{2\left(1-\cos \theta \right)}\end{array}$

コード(Emacs)

Python 3

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

from sympy import symbols, pprint, conjugate, expand, cos, sin, pi, summation

print('1.')
a, b, c, d = symbols('a b c d', real=True)
alpha = a + b * 1J
pprint(alpha)
pprint(conjugate(alpha))
pprint(-conjugate(alpha))
pprint(expand(1j * conjugate(alpha)))
pprint(expand(-1j * conjugate(alpha)))

print('5')
h = symbols('h', integer=True)
n, i = symbols('n i', integer=True)
w = cos(2 * pi / n) + 1j * sin(2 * pi / n)

pprint(w)
s = summation(w ** (i * h), (i, 0, n - 1))
pprint(s)

print('6.')
theta = symbols('θ', real=True)
pprint(summation(cos(i * theta), (i, 0, n)))
pprint(summation(sin(i * theta), (i, 1, n)))

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
a + 1.0⋅ⅈ⋅b
a - 1.0⋅ⅈ⋅b
-a + 1.0⋅ⅈ⋅b
1.0⋅ⅈ⋅a + 1.0⋅b
-1.0⋅ⅈ⋅a - 1.0⋅b
5
         ⎛2⋅π⎞      ⎛2⋅π⎞
1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟
         ⎝ n ⎠      ⎝ n ⎠
⎧                                                                     h    
⎪                                          ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞     
⎪                 n                    for ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟  = 1
⎪                                          ⎝         ⎝ n ⎠      ⎝ n ⎠⎠     
⎪                                                                          
⎪                             h⋅n                                          
⎪  ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞                                             
⎨- ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟    + 1                                      
⎪  ⎝         ⎝ n ⎠      ⎝ n ⎠⎠                                             
⎪────────────────────────────────────               otherwise              
⎪                              h                                           
⎪   ⎛         ⎛2⋅π⎞      ⎛2⋅π⎞⎞                                            
⎪ - ⎜1.0⋅ⅈ⋅sin⎜───⎟ + cos⎜───⎟⎟  + 1                                       
⎪   ⎝         ⎝ n ⎠      ⎝ n ⎠⎠                                            
⎩                                                                          
6.
  n           
 ___          
 ╲            
  ╲   cos(i⋅θ)
  ╱           
 ╱            
 ‾‾‾          
i = 0         
  n           
 ___          
 ╲            
  ╲   sin(i⋅θ)
  ╱           
 ╱            
 ‾‾‾          
i = 1
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="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>
α = <input id="a0" type="number" value="2"> + <input id="b0" type="number" value="3">i
<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_a = document.querySelector('#a0'),
    input_b = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a, input_b],
    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),
        a = parseFloat(input_a.value),
        b = parseFloat(input_b.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [[a, b, 'red'],
                  [-a, b, 'green'],
                  [a, -b, 'blue'],
                  [b, a, 'orange'],
                  [-b, -a, 'brown']];
       
    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('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]);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .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', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
α = + i