偏角・対数の極限
偏角・対数の極限
(1)
\[ \lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\left(-\alpha\right)-\pi & x\rightarrow-0 \end{cases} \](2)
\[ \lim_{x\rightarrow\pm0}\left(\Log\left(\alpha x\right)-\Log\left(x\right)\right)=\begin{cases} \Log\alpha & x\rightarrow+0\\ \Log\left(-\alpha\right)-\pi i & x\rightarrow-0 \end{cases} \](1)
\begin{align*} \lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} & =\lim_{x\rightarrow\pm0}\left\{ \Arg\left(\frac{\alpha x}{x}\right)+2\pi\mzp_{-1,0}\left(-\pi,\pi;\Arg\left(\alpha x\right)+\Arg x^{-1}\right)-2\pi\delta_{\pi,\Arg\left(x\right)}\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha\right)+2\pi\mzp_{-1,0}\left(-\pi,\pi;\Arg\left(\alpha x\right)+\Arg x^{-1}\right)-2\pi\delta_{\pi,\Arg\left(x\right)}\right\} \\ & =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\alpha+2\pi H_{0}\left(\Arg\left(-\alpha\right)\right)-2\pi & x\rightarrow-0 \end{cases}\\ & =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\alpha+\pi-\Arg\alpha+\Arg\left(-\alpha\right)-2\pi & x\rightarrow-0 \end{cases}\\ & =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\left(-\alpha\right)-\pi & x\rightarrow-0 \end{cases} \end{align*}(1)-2
\begin{align*} \lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} & =\lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha\frac{x}{\left|x\right|}\right)-\Arg\left(\frac{x}{\left|x\right|}\right)\right\} \\ & =\Arg\left(\pm\alpha\right)-\Arg\left(\pm1\right)\\ & =\begin{cases} \Arg\alpha & x\rightarrow+0\\ \Arg\left(-\alpha\right)-\pi & x\rightarrow-0 \end{cases} \end{align*}(2)
\begin{align*} \lim_{x\rightarrow\pm0}\left(\Log\left(\alpha x\right)-\Log\left(x\right)\right) & =\lim_{x\rightarrow\pm0}\left(\Log\left|\alpha x\right|+i\Arg\left(\alpha x\right)-\Log\left|x\right|-i\Arg\left(x\right)\right)\\ & =\lim_{x\rightarrow\pm0}\left(\Log\left|\alpha x\right|-\Log\left|x\right|+i\left(\Arg\left(\alpha x\right)-\Arg\left(x\right)\right)\right)\\ & =\lim_{x\rightarrow\pm0}\left(\Log\frac{\left|\alpha x\right|}{\left|x\right|}+i\left(\Arg\left(\alpha x\right)-\Arg\left(x\right)\right)\right)\\ & =\lim_{x\rightarrow\pm0}\left(\Log\left|\alpha\right|+i\left(\Arg\left(\alpha x\right)-\Arg\left(x\right)\right)\right)\\ & =\begin{cases} \Log\left|\alpha\right|+i\Arg\alpha & x\rightarrow+0\\ \Log\left|\alpha\right|+i\left(\Arg\left(-\alpha\right)-\pi\right) & x\rightarrow-0 \end{cases}\\ & =\begin{cases} \Log\alpha & x\rightarrow+0\\ \Log\left(-\alpha\right)-\pi i & x\rightarrow-0 \end{cases} \end{align*}
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偏角・対数と符号関数の関係
\[
\Arg\left(z\right)=-i\Log\left(\sgn\left(z\right)\right)
\]
絶対値の冪乗
\[
\left(\left|\alpha\right|\beta\right)^{\gamma}=\left|\alpha\right|^{\gamma}\beta^{\gamma}
\]
0の極限のべき乗と0の極限乗
\[
\lim_{z\rightarrow0}z^{\alpha}=\begin{cases}
0 & 0<\Re\left(\alpha\right)\\
1 & \alpha=0\\
\text{発散} & \Re\left(\alpha\right)<0\lor\left(\Re\left(\alpha\right)=0\land\Im\left(\alpha\right)\ne0\right)
\end{cases}
\]
冪乗の性質
\[
\pv\alpha^{\beta}\pv\alpha^{\gamma}=\pv\alpha^{\beta+\gamma}
\]