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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複素指数関数の極形式
\[
\alpha^{\beta}=\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\arg\alpha}e^{i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha\right)}
\]
絶対値の冪乗
\[
\left(\left|\alpha\right|\beta\right)^{\gamma}=\left|\alpha\right|^{\gamma}\beta^{\gamma}
\]
負数の偏角と対数
\[
\Arg\alpha-\Arg\left(-\alpha\right)=2\pi H_{0}\left(\Arg\left(\alpha\right)\right)-\pi
\]
複素共役の偏角と対数
\[
\Arg\overline{z}=-\Arg z+2\pi\delta_{\pi,\Arg z}
\]