絶対値の冪乗
絶対値の冪乗
\[ \left|\alpha^{b}\right|=\left|\alpha\right|^{b} \]
\begin{align*} \left(\left|\alpha\right|^{b}\right)^{\gamma} & =\left|\alpha\right|^{b\gamma} \end{align*}
(1)
\[ \left(\left|\alpha\right|\beta\right)^{\gamma}=\left|\alpha\right|^{\gamma}\beta^{\gamma} \](2)
\[ \alpha^{\beta}=\left|\alpha\right|^{\beta}\sgn^{\beta}\left(\alpha\right) \](3)
\(b\in\mathbb{R}\)とする。\[ \left|\alpha^{b}\right|=\left|\alpha\right|^{b} \]
(4)
\(b\in\mathbb{R}\)とする。\begin{align*} \left(\left|\alpha\right|^{b}\right)^{\gamma} & =\left|\alpha\right|^{b\gamma} \end{align*}
(1)
\begin{align*} \left(\left|\alpha\right|\beta\right)^{\gamma} & =e^{\gamma\Log\left(\left|\alpha\right|\beta\right)}\\ & =e^{\gamma\left(\ln\left|\alpha\right|+\Log\beta\right)}\\ & =\left|\alpha\right|^{\gamma}\beta^{\gamma} \end{align*}(2)
\begin{align*} \alpha^{\beta} & =\left(\left|\alpha\right|\sgn\alpha\right)^{\beta}\\ & =\left|\alpha\right|^{\beta}\sgn^{\beta}\left(\alpha\right) \end{align*}(3)
\[ \begin{align*}\left|\alpha^{b}\right| & =\left|e^{b\Log\alpha}\right|\\ & =\left|e^{b\left(\Log\left|\alpha\right|+i\Arg\left(\alpha\right)\right)}\right|\\ & =\left|\left|\alpha\right|^{b}e^{ib\arg\alpha}\right|\\ & =\left|\left|\alpha\right|^{b}\right|\left|e^{ib\arg\alpha}\right|\\ & =\left|\alpha\right|^{b} \end{align*} \](4)
\begin{align*} \left(\left|\alpha\right|^{b}\right)^{\gamma} & =e^{\gamma\Log\left|\alpha\right|^{b}}\\ & =e^{\gamma b\ln\left|\alpha\right|}\\ & =\left|\alpha\right|^{b\gamma} \end{align*}
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複素数の冪関数の定義
\[
\alpha^{\beta}=e^{\beta\log\alpha}
\]
積が非負実数のべき乗
\[
\left(\Arg\left(\alpha\right)\ne\pi\lor\Arg\left(\beta\right)\ne\pi\right)\land0\leq a\beta\rightarrow\left(\alpha\beta\right)^{\gamma}=\alpha^{\gamma}\beta^{\gamma}
\]
偏角・対数と絶対値
\[
\Log\left(\left|\alpha\right|\beta\right)=\ln\left|\alpha\right|+\Log\beta
\]
偏角の和と積の偏角
\[
\Arg\left(\alpha\right)+\Arg\left(\beta\right)=?\Arg\left(\alpha\beta\right)
\]