複素数と複素共役の和・差
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*}
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偏角・対数と符号関数の関係
\[
\Arg\left(z\right)=-i\Log\left(\sgn\left(z\right)\right)
\]
eの冪乗の基本
\[
e^{\alpha+\beta}=e^{\alpha}e^{\beta}
\]
2乗のルート
\[
\sqrt{\alpha^{2}}=\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)}
\]
偏角・対数と絶対値
\[
\Log\left(\left|\alpha\right|\beta\right)=\ln\left|\alpha\right|+\Log\beta
\]