数学 - 線形代数学 - R^nにおけるベクトル – 複素数(オイラーの公式、三角関数(正弦関数、余弦関数、加法定理))

ラング線形代数学(上)
原著

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

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の1章(R^nにおけるベクトル)、6(複素数)、練習問題7.を取り組んでみる。

7. 1. 
      $\begin{array}{l}{e}^{i\left({\theta }_1+{\theta }_2\right)}=\cos \left({\theta }_1+{\theta }_2\right)+i\sin \left({\theta }_1+{\theta }_2\right)\\ =\cos {\theta }_1\cos {\theta }_2-\sin {\theta }_1\sin {\theta }_2+i\sin {\theta }_1\cos {\theta }_2+i\cos {\theta }_1\sin {\theta }_2\\ =\left(\cos {\theta }_1+i\sin {\theta }_1\right)\left(\cos {\theta }_2+i\sin {\theta }_2\right)\\ =e^{i{\theta }_1}·e^{i{\theta }_2}\\ \\ a+bi\\ \sqrt{a^2+b^2}=1\\ 1\sin \theta +i1\cos \theta =e^{it}\end{array}$
   2. 
      $\begin{array}{l}z=a+bi\\ z=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{a^2+b^2}}i\right)\\ =\sqrt{a^2+b^2}\left(\cos \theta +i\sin \theta \right)\end{array}$
   3. 
      $\begin{array}{l}{z}_1{z}_2\\ =r_1{e}^{i{\theta }_1}{r}_2{e}^{i{\theta }_2}\\ =r_1{r}_2{e}^{i{\theta }_1}{e}^{i{\theta }_2}\\ =r_1{r}_2{e}^{i\left({\theta }_1+{\theta }_2\right)}\end{array}$
   4. 
      $\begin{array}{l}z=r{e}^{i\theta }\\ z^{\frac{1}{n}}=r^{\frac{1}{n}}{e}^{\frac{i\theta }{n}}\\ w=r^{\frac{1}{n}}{e}^{i\frac{\theta }{n}}\\ w^n=r{e}^{in\theta ·\frac{1}{n}}=r{e}^{i\theta }\end{array}$

コード(Emacs)

HTML5

<pre id="output0"></pre>
θ = <input id="theta0" type="number" value="1.2">
θ_1 = <input id="theta1" type="number" value="2.3">
θ_2 = <input id="theta2" type="number" value="3.4">
<br>
z = <input id="a0" type="number" value="5.6"> + <input id="b0" type="number" value="7.8">i
<br>
r_1 = <input id="r1" type="number" min="0" value="9.0">
r_2 = <input id="r2" type="number" min="0" value="1.2">
<br>
n = <input id="n0" type="number" min="1" step="1" value="5">
<br>
<button id="run0">run</button>
<button id="clear0">clear</button>

<script src="sample7.js"></script>

JavaScript

let pre0 = document.querySelector('#output0'),
    input_0 = document.querySelector('#theta0'),
    input_1 = document.querySelector('#theta1'),
    input_2 = document.querySelector('#theta2'),
    input_a = document.querySelector('#a0'),
    input_b = document.querySelector('#b0'),
    input_r1 = document.querySelector('#r1'),
    input_r2 = document.querySelector('#r2'),
    input_n = document.querySelector('#n0'),
    inputs = [input_0, input_1, input_2, input_a, input_b, input_r1, input_r2,
             input_n],
    btn0 = document.querySelector('#run0'),
    btn1 = document.querySelector('#clear0'),
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let result = [];

        for (let i = start; i < end; i += step) {
            result.push(i);
        }
        return result;
    };

let Complex = (x, y) => {
    let that = {},
        toString = () => `${x} + ${y}i`,
        real = () => x,
        imag = () => y,
        add = (z) => Complex(x + z.real(), y + z.imag()),
        mul = (z) => Complex(x * z.real() - y * z.imag(),
                             x * z.imag() + y * z.real()),
        conjugate = () => Complex(x, -y),
        inv = () => {
            let den = Math.pow(x, 2) + Math.pow(y, 2);
            return Complex(x / den, -y / den);
        },
        mag = () => Math.sqrt(Math.pow(x, 2) + Math.pow(y, 2)),
        isEqual = (z) => x === z.real() && y === z.imag();

    that.toString = toString;
    that.real = real;
    that.imag = imag;
    that.add = add;
    that.mul = mul;
    that.conjugate = conjugate;
    that.inv = inv;
    that.mag = mag;
    that.isEqual = isEqual;
    
    return that;
};

let f = (theta) => Complex(Math.cos(theta), Math.sin(theta));

let output = () => {
    let theta = parseFloat(input_0.value),
        theta1 = parseFloat(input_1.value),
        theta2 = parseFloat(input_2.value);
    

    p('a.');
    let left= f(theta1 + theta2),
        right = f(theta1).mul(f(theta2));

    p(left.isEqual(right));
    p(left);
    p(right);

    p('b.');
    let a = parseFloat(input_a.value),
        b = parseFloat(input_b.value),
        r = Math.sqrt(Math.pow(a, 2) + Math.pow(b, 2)),
        theta_b = Math.acos(a / r);

    p(`r = ${r}, θ = ${theta_b}`);
    left = Complex(a, b);
    right = Complex(r, 0).mul(f(theta_b));
    p(left.isEqual(right));
    p(left);
    p(right);

    p('c.');
    let r1 = parseFloat(input_r1.value),
        r2 = parseFloat(input_r2.value),
        z1 = Complex(r1, 0).mul(f(theta1)),
        z2 = Complex(r2, 0).mul(f(theta2));

    left = z1.mul(z2);
    right = Complex(r1 * r2, 0).mul(f(theta1 + theta2));
    p(left.isEqual(right));
    p(left);
    p(right);

    p('d.');
    let z = Complex(a, b),
        n = parseInt(input_n.value, 10),
        w = Complex(Math.pow(z.mag(), 1 / n), 0).mul(f(Math.acos(a / z.mag()) / n));

    let wn = w;
    for (let i = 2; i <= n; i += 1) {
        wn = wn.mul(w);
    }
    p(wn.isEqual(z));
    p(wn);
    p(z);
};

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

output();

θ = θ_1 = θ_2 =
z = + i
r_1 = r_2 =
n =