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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2乗のルート
\[
\sqrt{\alpha^{2}}=\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)}
\]
対数と偏角の性質
\[
\log\alpha^{\beta}=\beta\log\alpha+\log1
\]
逆数の偏角と対数
\[
\Arg z^{-1}=-\Arg z+2\pi\delta_{\pi,\Arg\left(z\right)}
\]
偏角・対数と絶対値
\[
\Log\left(\left|\alpha\right|\beta\right)=\ln\left|\alpha\right|+\Log\beta
\]