数学 – JavaScript - 新しい数とその表示 - 複素数と複素平面 – 複素平面(複素数の極形式)

数学読本〈3〉

学習環境/開発環境

• 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
• OS X El Capitan - Apple (OS)
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• kjs-math-number (JavaScript Library)
• Safari(Web browser)

数学読本〈3〉平面上のベクトル/複素数と複素平面/空間図形/2次曲線/数列 (松坂 和夫(著)、岩波書店)の第10章(新しい数とその表示 - 複素数と複素平面)、10.1(複素平面)、複素数の極形式、問7、8、9.を取り組んでみる。

問7.

1. 
   $\sqrt{2}\left(\cos \frac{7\pi }{4}+\sin \frac{7\pi }{4}i\right)$
2. 
   $\sqrt{2}\left(\cos \frac{5\pi }{4}+\sin \frac{5\pi }{4}i\right)$
3. 
   $2\left(\cos \frac{\pi }{3}+\sin \frac{\pi }{3}i\right)$
4. 
   $2\left(\cos \frac{5\pi }{6}+\sin \frac{5\pi }{6}i\right)$
5. 
   $\cos \frac{\pi }{2}+\sin \frac{\pi }{2}i$
6. 
   $2\left(\cos \frac{3\pi }{2}+\sin \frac{3\pi }{2}i\right)$
7. 
   $3\left(\cos \pi +\sin \pi \right)$
8. 
   $\cos \frac{7\pi }{4}+\sin \frac{7\pi }{4}i$

問8.

$\begin{array}{l}\left|i\overline{z}\right|=\left|i\right|\left|\overline{z}\right|=\left|z\right|\\ \\ \arg \left(i\overline{z}\right)=\arg \left(i\right)+\arg \left(\overline{z}\right)\\ =\frac{\pi }{2}-\arg \left(z\right)\end{array}$

問9.

1. 
   $\begin{array}{l}\left|z+\frac{1}{\overline{z}}\right|=\left|\frac{z\overline{z}+1}{\overline{z}}\right|=\frac{r^2+1}{r}\\ \arg \left(z+\frac{1}{\overline{z}}\right)=\arg \left(\frac{z\overline{z}+1}{\overline{z}}\right)\\ =\arg \left(z\overline{z}+1\right)-\arg \left(\overline{z}\right)\\ =\arg \left(r^2+1\right)+\arg \left(z\right)\\ =\theta \end{array}$
2. 
   $\begin{array}{l}\left|r+z\right|=\left|r+r\left(\cos \theta +i\sin \theta \right)\right|\\ =r\left|1+\cos \theta +i\sin \theta \right|\\ =r\left|1+{\cos }^2\frac{\theta }{2}-{\sin }^2\frac{\theta }{2}+i2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right|\\ =r\left|1+{\cos }^2\frac{\theta }{2}-1+{\cos }^2\frac{\theta }{2}+i2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right|\\ =2r\left|{\cos }^2\frac{\theta }{2}+i\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right|\\ =2r\cos \frac{\theta }{2}\left|\cos \frac{\theta }{2}+i\sin \frac{\theta }{2}\right|\\ =2r\cos \frac{\theta }{2}\\ \arg \left(r+z\right)=\frac{\theta }{2}\end{array}$
3. 
   $\begin{array}{l}r-z=r-r\left(\cos \theta +i\sin \theta \right)\\ =r\left(1-\cos \theta -i\sin \theta \right)\\ =r\left(1-{\cos }^2\frac{\theta }{2}+{\sin }^2\frac{\theta }{2}-2i\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right)\\ =r\left(1-1+{\sin }^2\frac{\theta }{2}+{\sin }^2\frac{\theta }{2}-2i\sin \frac{\theta }{2}\cos \frac{\theta }{2}\right)\\ =2r\sin \frac{\theta }{2}\left(\sin \frac{\theta }{2}-i\cos \frac{\theta }{2}\right)\\ =2r\sin \frac{\theta }{2}\left(\cos \left(\frac{\pi }{2}-\frac{\theta }{2}\right)-i\sin \left(\frac{\pi }{2}-\frac{\theta }{2}\right)\right)\\ =2r\sin \frac{\theta }{2}\left(\cos \left(\frac{\theta }{2}-\frac{\pi }{2}\right)+i\sin \left(\frac{\theta }{2}-\frac{\pi }{2}\right)\right)\\ \left|r-z\right|=2r\sin \frac{\theta }{2}\\ \arg \left(r-z\right)=\frac{\theta }{2}-\frac{\pi }{2}\end{array}$
4. 
   $\begin{array}{l}\left|\frac{r-z}{r+z}\right|=\frac{\left|r-z\right|}{\left|r+z\right|}=\frac{2r\sin \frac{\theta }{2}}{2r\cos \frac{\theta }{2}}=\tan \frac{\theta }{2}\\ \arg \left(\frac{r-z}{r+z}\right)=\arg \left(r-z\right)-\arg \left(r+z\right)\\ =\frac{\theta }{2}-\frac{\pi }{2}-\frac{\theta }{2}=-\frac{\pi }{2}\end{array}$

number.js で確認。

JavaScript

コード(Emacs)

(function () {
    'use strict';
    var div_output = document.querySelector('#output0'),
        button_calc = document.querySelector('#calc0'),
        nl = '<br><br>',
        calc;

    calc = function () {
        var z = new Complex(Math.floor(Math.random() * 10),
                            Math.floor(Math.random() * 10)),
            r = z.magnitude(),
            theta = z.angle(),
            z0,
            output;

        output = 'z = <math>' + z + '</math>, r = ' + r + ', θ = ' + theta + nl;
        [z.add((1).div(z.conjugate())),
         r.add(z),
         r.sub(z),
         r.sub(z).div(r.add(z))].forEach(function (z0, i) {
             output +=
                 '(' + (i + 1) + ') r = ' + z0.magnitude() +
                 ', θ = ' + z0.angle() + nl;
         });
        div_output.innerHTML = output;
    };

    calc();

    button_calc.onclick = calc;
}());