カテゴリー: 複素数

\[ \lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\left(-\alpha\right)-\pi & x\rightarrow-0 \end{cases} \]

\[ \Arg\alpha-\Arg\left(-\alpha\right)=2\pi H_{0}\left(\Arg\left(\alpha\right)\right)-\pi \]

\[ \Log\alpha^{\beta}=\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)+\mod\left(\Re\left(\beta\right)\Arg\left(\alpha\right)+\Im\left(\beta\right)\ln\left|\alpha\right|,-2\pi,\pi\right) \]

\[ \sqrt{\alpha^{2}}=\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \]

\[ \Arg\left(z\right)=-i\Log\left(\sgn\left(z\right)\right) \]

\[ \Log\sgn\alpha=i\Arg\alpha \]

\[ \left(\left|\alpha\right|\beta\right)^{\gamma}=\left|\alpha\right|^{\gamma}\beta^{\gamma} \]

\[ \Log\left(\left|\alpha\right|\beta\right)=\ln\left|\alpha\right|+\Log\beta \]

\[ \Arg\overline{z}=-\Arg z+2\pi\delta_{\pi,\Arg z} \]

\[ \Arg\alpha+\Arg\beta=\Arg\left(\alpha\beta\right)+2\pi\mzp_{-1,0}\left(-\pi,\pi;\Arg\alpha+\Arg\beta\right) \]