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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| タイトル | 偏角・対数の極限 |
| URL | https://www.nomuramath.com/zxz1pmio/ |
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0の極限のべき乗と0の極限乗
\[
\lim_{z\rightarrow0}z^{\alpha}=\begin{cases}
0 & 0<\Re\left(\alpha\right)\\
1 & \alpha=0\\
\text{発散} & \Re\left(\alpha\right)<0\lor\left(\Re\left(\alpha\right)=0\land\Im\left(\alpha\right)\ne0\right)
\end{cases}
\]
積が非負実数のべき乗
\[
\left(\Arg\left(\alpha\right)\ne\pi\lor\Arg\left(\beta\right)\ne\pi\right)\land0\leq a\beta\rightarrow\left(\alpha\beta\right)^{\gamma}=\alpha^{\gamma}\beta^{\gamma}
\]
指数関数の実部と虚部
\[
\left|\alpha^{\beta}\right|=\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\left(\alpha\right)}
\]
対数の指数exp(Log(z))と指数の対数Log(exp(z))の違い
\[
\Re\left(z\right)+i\mod\left(\Im\left(z\right),-2\pi,\pi\right)=\Log\left(\exp\left(z\right)\right)
\]