絶対値の冪乗

by nomura · 2021年8月30日

絶対値の冪乗

(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*}