対数と偏角の性質
対数と偏角の主値の性質
\(n\in\mathbb{Z}\)とする。
\(n\in\mathbb{Z}\)とする。
(1)
\[ \Arg\left(\alpha\beta\right)\overset{2\pi}{\equiv}\Arg\left(\alpha\right)+\Arg\left(\beta\right) \](2)
\[ \Arg\left(\pv\left(\alpha^{\beta}\right)\right)\overset{2\pi}{\equiv}\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right) \](3)
\[ \Arg\left(\alpha^{n}\right)\overset{2\pi}{\equiv}n\Arg\left(\alpha\right) \](4)
\[ \Arg\left(\frac{1}{\alpha}\right)\overset{2\pi}{\equiv}-\Arg\left(\alpha\right) \](5)
\[ \begin{align*}\Log\left(\alpha\beta\right) & \overset{2\pi}{\equiv}\Log\alpha+\Log\beta\end{align*} \](6)
\[ \begin{align*}\Log\left(\pv\left(\alpha^{\beta}\right)\right) & \overset{2\pi i}{\equiv}\beta\Log\alpha\end{align*} \](7)
\[ \Log\alpha^{n}\overset{2\pi i}{\equiv}n\Log\alpha \](8)
\[ \Log\frac{1}{\alpha}\overset{2\pi i}{\equiv}-\Log\alpha \]等号が成り立たない例
(1)
\(\alpha=\beta=-1\)とすると左辺は\(\Arg\left(\left(-1\right)\left(-1\right)\right)=\Arg\left(1\right)=0\)で右辺は\(\Arg\left(-1\right)+\Arg\left(-1\right)=\pi+\pi=2\pi\)となるので等号は成り立たない。(2)
\(\alpha=-1,\beta=2\)とすると左辺は\(\Arg\left(\pv\left(\left(-1\right)^{2}\right)\right)=\Arg\left(1\right)=0\)で右辺は\(\Im\left(2\right)\ln\left|-1\right|+\Re\left(2\right)\Arg\left(-1\right)=2\pi\)となるので等号は成り立たない。(3)
\(\alpha=-1,n=2\)とすると左辺は\(\Arg\left(\left(-1\right)^{2}\right)=\Arg\left(1\right)=0\)で右辺は\(2\Arg\left(-1\right)=2\pi\)となるので等号は成り立たない。(4)
\(\alpha=-1\)とすると左辺は\(\Arg\left(\frac{1}{-1}\right)=\Arg\left(-1\right)=\pi\)で右辺は\(-\Arg\left(-1\right)=-\pi\)となるので等号は成り立たない。(5)
\(\alpha=\beta=-1\)とすると左辺は\(\begin{align*}\Log\left(\left(-1\right)\left(-1\right)\right) & =\Log\left(1\right)=0\end{align*} \)で右辺は\(\Log\left(-1\right)+\Log\left(-1\right)=\pi i+\pi i=2\pi i\)となるので等号は成り立たない。(6)
\(\alpha=-1,\beta=2\)とすると左辺は\(\Log\left(\pv\left(\left(-1\right)^{2}\right)\right)=\Log\left(1\right)=0\)で右辺は\(2\Log\left(-1\right)=2\pi i\)となるので等号は成り立たない。(7)
\(\alpha=-1,n=2\)とすると左辺は\(\Log\left(\left(-1\right)^{2}\right)=\Log\left(1\right)=0\)で右辺は\(2\Log\left(-1\right)=2\pi i\)となるので等号は成り立たない。(8)
\(\alpha=-1\)とすると左辺は\(\Log\frac{1}{-1}=\Log\left(-1\right)=\pi i\)で右辺は\(-\Log\left(-1\right)=-\pi i\)となるので等号は成り立たない。(1)
\begin{align*} \Arg\left(\alpha\beta\right) & =\Arg\left(\left|\alpha\right|e^{i\Arg\left(\alpha\right)}\left|\beta\right|e^{i\Arg\left(\beta\right)}\right)\\ & =\Arg\left(\left|\alpha\beta\right|e^{i\left(\Arg\left(\alpha\right)+\Arg\left(\beta\right)\right)}\right)\\ & \overset{2\pi}{\equiv}\Arg\left(\alpha\right)+\Arg\left(\beta\right) \end{align*}(2)
\begin{align*} \Arg\left(\pv\left(\alpha^{\beta}\right)\right) & =\Arg\left(e^{\beta\Log\alpha}\right)\\ & =\Arg\left(e^{\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\left(\ln\left|\alpha\right|+i\Arg\left(\alpha\right)\right)}\right)\\ & =\Arg\left(e^{\left(\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)\right)+i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right)\right)}\right)\\ & =\Arg\left(\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\left(\alpha\right)}e^{i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right)\right)}\right)\\ & \overset{2\pi}{\equiv}\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right) \end{align*}(3)
(2)より、\begin{align*} \Arg\left(\alpha^{n}\right) & \overset{2\pi}{\equiv}n\Arg\left(\alpha\right) \end{align*}
(4)
(2)より、\begin{align*} \Arg\left(\frac{1}{\alpha}\right) & \overset{2\pi}{\equiv}-\Arg\left(\alpha\right) \end{align*}
(5)
\[ \begin{align*}\Log\left(\alpha\beta\right) & =\ln\left|\alpha\beta\right|+i\Arg\left(\alpha\beta\right)\\ & \overset{2\pi}{\equiv}\ln\left|\alpha\right|+\ln\left|\beta\right|+i\Arg\alpha+i\Arg\beta\\ & =\Log\alpha+\Log\beta \end{align*} \](6)
\[ \begin{align*}\Log\left(\pv\left(\alpha^{\beta}\right)\right) & =\Log e^{\beta\Log\alpha}\\ & =\Log e^{\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\left(\ln\left|\alpha\right|+i\Arg\alpha\right)}\\ & =\Log e^{\left(\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\alpha\right)+i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)}\\ & =\Log\left(\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\alpha}e^{i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)}\right)\\ & =\Log\left(\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\alpha}\right)+\Log e^{i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)}\\ & \overset{2\pi i}{\equiv}\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\alpha+i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)\\ & =\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\ln\left|\alpha\right|+i\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\Arg\alpha\\ & =\beta\left(\ln\left|\alpha\right|+i\Arg\alpha\right)\\ & =\beta\Log\alpha \end{align*} \](7)
(6)より、\[ \Log\alpha^{n}\overset{2\pi i}{\equiv}n\Log\alpha \]
(8)
(6)より、\[ \Log\frac{1}{\alpha}\overset{2\pi i}{\equiv}-\Log\alpha \]
対数と偏角の多価関数の性質
\(n\in\mathbb{Z}\)とする。
\(n\in\mathbb{Z}\)とする。
(1)
\[ \arg(\alpha\beta)=\arg(\alpha)+\arg(\beta) \](2)
\[ \arg\mv\alpha^{\beta}=\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha+\arg1 \](3)
\[ \arg\alpha^{n}=n\arg\alpha+\arg1 \](4)
\[ \arg(1/\alpha)=-\arg(\alpha) \](5)
\[ \log\left(\alpha\beta\right)=\log\alpha+\log\beta \](6)
\[ \log\mv\alpha^{\beta}=\beta\log\alpha+\log1 \](7)
\[ \log\alpha^{n}=n\log\alpha+\log1 \](8)
\[ \log\left(1/z\right)=-\log z \](1)
\begin{align*} \arg\left(\alpha\beta\right) & =\Arg\left(\alpha\beta\right)+\arg1\\ & =\Arg\alpha+\Arg\beta+\arg1\\ & =\arg\alpha+\arg\beta \end{align*}(1)-2
\begin{align*} \arg\left(\alpha\beta\right) & =\arg\left(\left|\alpha\right|e^{i\arg\alpha}\left|\beta\right|e^{i\arg\beta}\right)\\ & =\arg\left(\left|\alpha\beta\right|e^{i\left(\arg\alpha+\arg\beta\right)}\right)\\ & =\arg\alpha+\arg\beta \end{align*}(2)
\begin{align*} \arg\mv\alpha^{\beta} & =\Arg\left(\alpha^{\beta}\mv\left(1^{\beta}\right)\right)+\arg1\\ & =\Arg\alpha^{\beta}+\Arg\mv1^{\beta}+\arg1\\ & =\Arg\alpha^{\beta}+\Re\left(\beta\right)\arg1+\arg1\\ & =\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right)+\Re\left(\beta\right)\arg1+\arg1\\ & =\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\left(\alpha\right)+\arg1 \end{align*}(2)-2
\begin{align*} \arg\mv\alpha^{\beta} & =\arg e^{\beta\log\alpha}\\ & =\arg e^{\left(\Re\left(\beta\right)+\Im\left(\beta\right)i\right)\left(\ln\left|\alpha\right|+i\arg\alpha\right)}\\ & =\arg 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)}\\ & =\arg\left(\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)}\right)\\ & =\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha+\arg1 \end{align*}(3)
(2)より、\begin{align*} \arg\alpha^{n} & =\Im\left(n\right)\ln\left|\alpha\right|+\Re\left(n\right)\arg\alpha+\arg1\\ & =n\arg\alpha+\arg1 \end{align*}
(3)-2
\begin{align*} \arg\alpha^{n} & =\sum_{k=1}^{n}\arg\alpha\\ & =\sum_{k=1}^{n}\left\{ \Arg\alpha+2\pi m;m\in\mathbb{Z}\right\} \\ & =\left\{ n\Arg\alpha+2\pi\left(m_{1}+m_{2}+\cdots\cdots+m_{n}\right);m_{i}\in\mathbb{Z}\right\} \\ & =\left\{ n\Arg\alpha+2\pi m;m\in\mathbb{Z}\right\} \\ & =n\Arg\alpha+\left\{ 2\pi m;m\in\mathbb{Z}\right\} \\ & =n\Arg\alpha+\arg1\\ & =n\Arg\alpha+n\arg1+\arg1\\ & =n\arg\alpha+\arg1 \end{align*}(4)
(2)より、\begin{align*} \arg\left(1/\alpha\right) & =\arg\alpha^{-1}\\ & =-\arg\alpha+\arg1\\ & =-\arg\alpha \end{align*}
(5)
\begin{align*} \log\left(\alpha\beta\right) & =\Log\left(\alpha\beta\right)+\log1\\ & =\Log\alpha+\Log\beta+\log1\\ & =\log\alpha+\log\beta \end{align*}(5)-2
\begin{align*} \log\left(\alpha\beta\right) & =\ln\left|\alpha\beta\right|+i\arg\left(\alpha\beta\right)\\ & =\ln\left|\alpha\right|+\ln\left|\beta\right|+i\arg\alpha+i\arg\beta\\ & =\log\alpha+\log\beta \end{align*}(6)
\begin{align*} \log\mv\left(\alpha^{\beta}\right) & =\Log\left(\alpha^{\beta}\mv\left(1^{\beta}\right)\right)+\log1\\ & =\Log\alpha^{\beta}+\Log\mv\left(1^{\beta}\right)+\log1\\ & =\Log\alpha^{\beta}+\Log e^{\beta\log1}+\log1\\ & =\beta\Log\alpha+\beta\log1+\log1\\ & =\beta\left(\Log\alpha+\log1\right)+\log1\\ & =\beta\log\alpha+\log1 \end{align*}(6)
\begin{align*} \log\mv\alpha^{\beta} & =\log e^{\beta\log\alpha}\\ & =\log e^{\left(\Re\left(\beta\right)+\Im\left(\beta\right)i\right)\left(\ln\left|\alpha\right|+i\arg\alpha\right)}\\ & =\log 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)}\\ & =\log\left(\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)}\right)\\ & =\ln\left(\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\arg\alpha}\right)+\log e^{i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha\right)}\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\arg\alpha+i\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha\right)+\log1\\ & =\left(\Re\left(\beta\right)+\Im\left(\beta\right)i\right)\ln\left|\alpha\right|+i\left(\Re\left(\beta\right)+\Im\left(\beta\right)i\right)\arg\alpha+\log1\\ & =\beta\left(\ln\left|\alpha\right|+i\arg\alpha\right)+\log1\\ & =\beta\log\alpha+\log1 \end{align*}(7)
(6)より、\[ \log\alpha^{n}=n\log\alpha+\log1 \]
(7)-2
\begin{align*} \log\alpha^{n} & =\sum_{k=1}^{n}\log\alpha\\ & =\sum_{k=1}^{n}\left\{ \Log\alpha+2\pi im;m\in\mathbb{Z}\right\} \\ & =\left\{ n\Log\alpha+2\pi i\left(m_{1}+m_{2}+\cdots\cdots+m_{n}\right);m_{i}\in\mathbb{Z}\right\} \\ & =\left\{ n\Log\alpha+2\pi im;m\in\mathbb{Z}\right\} \\ & =n\Log\alpha+\left\{ 2\pi im;m\in\mathbb{Z}\right\} \\ & =n\Log\alpha+\log1\\ & =n\Log\alpha+n\log1+\log1\\ & =n\log\alpha+\log1 \end{align*}(8)
(6)より、\begin{align*} \log\left(1/\alpha\right) & =\log\alpha^{-1}\\ & =-\log\alpha+\log1\\ & =-\log\alpha \end{align*}
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偏角・対数と符号関数の関係
\[
\Arg\left(z\right)=-i\Log\left(\sgn\left(z\right)\right)
\]
複素数の冪関数の定義
\[
\alpha^{\beta}=e^{\beta\log\alpha}
\]
逆数の偏角と対数
\[
\Arg z^{-1}=-\Arg z+2\pi\delta_{\pi,\Arg\left(z\right)}
\]
負数の偏角と対数
\[
\Arg\alpha-\Arg\left(-\alpha\right)=2\pi H_{0}\left(\Arg\left(\alpha\right)\right)-\pi
\]