複素数

complex number

直交形式(rectangular form)

$ z = a + bi

※複素数の虚部は実数

極形式(polar form)

複素数を大きさと角度で表現できる。

複素数は二次元の複素平面上にあるので、極座標でも表現出来る

極形式を用いると複素数の累乗を比較的、簡単に求める事が出来る。

$ z = r\cos\phi + r\sin\phi i = r(\cos\phi + i\sin\phi) = re^{i\phi}

https://gyazo.com/0b73510d64791bdc7baf780bc809695f

$ r = |z| = \sqrt{a^2+b^2}

偏角の求め方

$ \tan(\phi) = \frac{b}{a}

$ \phi = \tan^{-1}\frac{b}{a}

複素数の絶対値

$ |z| = \sqrt{a^2+b^2}

複素数の積と商

$ z_1 = \textcolor{orange}{r_1}(\cos\textcolor{magenta}{\theta_1} +i \sin\textcolor{magenta}{\theta_1})

$ z_2 = \textcolor{orange}{r_2}(\cos\textcolor{magenta}{\theta_2} +i \sin\textcolor{magenta}{\theta_2)}

$ z_1z_2 = \textcolor{orange}{r_1r_2}(\cos(\textcolor{magenta}{\theta_1 + \theta_2}) + i\sin(\textcolor{magenta}{\theta_1 + \theta_2}))

$ \frac{z_1}{z_2} = \textcolor{orange}{\frac{r_1}{r_2}}(\cos(\textcolor{magenta}{\theta_1 - \theta_2}) + i\sin(\textcolor{magenta}{\theta_1 - \theta_2}))

複素数の冪乗

$ z_1 = \textcolor{orange}{r_1}(\cos\textcolor{magenta}{\theta_1} +i \sin\textcolor{magenta}{\theta_1})

$ (z_1)^n = (\textcolor{orange}{r_1})^n(\cos (n\cdot \textcolor{magenta}{\theta_1}) +i \sin (n\cdot \textcolor{magenta}{\theta_1}))

例題

$ z^7 = 128i

$ z^7 = r^7(\cos(7\cdot\theta) + i \sin(7\cdot\theta)) = 128(\cos(90\degree + k \cdot 360\degree) + i \sin(90\degree + k\cdot 360\degree)

$ r^7 = 128

$ r = 2

$ 7\cdot\theta = 90\degree + k \cdot 360\degree

$ \theta = \frac{90\degree}{7} + k \cdot \frac{360\degree}{7}