複素数と複素共役の和・差

by nomura · 2021年5月2日

複素数と複素共役の和・差

(1)

\[ z\pm\overline{z}=2H\left(\pm1\right)\Re z+2iH\left(\mp1\right)\Im z \]

(2)

\[ -z\pm\overline{z}=-2H\left(\mp1\right)\Re z-2iH\left(\pm1\right)\Im z \]

(3)

\[ \overline{z}\pm z=2H\left(\pm1\right)\Re z-2iH\left(\mp1\right)\Im z \]

(4)

\[ -\overline{z}\pm z=-2H\left(\mp1\right)\Re z+2iH\left(\pm1\right)\Im z \]

-

\(H\left(x\right)\)はヘヴィサイドの階段関数、\(\overline{z}\)は複素共役。

(1)

\begin{align*} z\pm\overline{z} & =\Re z+i\Im z\pm\left(\Re z-i\Im z\right)\\ & =\left(1\pm1\right)\Re z+i\left(1\mp1\right)\Im z\\ & =2H\left(\pm1\right)\Re z+2iH\left(\mp1\right)\Im z \end{align*}

(2)

\begin{align*} -z\pm\overline{z} & =-\left(z\mp\overline{z}\right)\\ & =-\left\{ 2H\left(\mp1\right)\Re z+2iH\left(\pm1\right)\Im z\right\} \\ & =-2H\left(\mp1\right)\Re z-2iH\left(\pm1\right)\Im z \end{align*}

(3)

\begin{align*} \overline{z}\pm z & =\pm\left(z\pm\overline{z}\right)\\ & =\pm\left(2H\left(\pm1\right)\Re z+2iH\left(\mp1\right)\Im z\right)\\ & =\pm2H\left(\pm1\right)\Re z\pm2iH\left(\mp1\right)\Im z\\ & =2H\left(\pm1\right)\Re z-2iH\left(\mp1\right)\Im z \end{align*}

(4)

\begin{align*} -\overline{z}\pm z & =-\left(\overline{z}\mp z\right)\\ & =-\left\{ 2H\left(\mp1\right)\Re z-2iH\left(\pm1\right)\Im z\right\} \\ & =-2H\left(\mp1\right)\Re z+2iH\left(\pm1\right)\Im z \end{align*}