複素指数関数の極形式

\begin{align*} \alpha^{\beta} & =e^{\beta\log\alpha}\\ & =e^{\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\left(\ln\left|\alpha\right|+i\arg\alpha\right)}\\ & =e^{\left(\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\arg\alpha\right)+i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha\right)}\\ & =\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)} \end{align*}