(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、
\[ \Log\frac{1}{\alpha}\overset{2\pi i}{\equiv}-\Log\alpha \]

(9)

\(\Rightarrow\)

\(\Log\alpha^{\beta}\)と\(\beta\Log\alpha\)は、
\begin{align*} \Log\alpha^{\beta} & =\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|+i\Arg\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)\\ & =\ln\left|e^{\Re\left(\beta\right)\ln\left|\alpha\right|}e^{-\Im\left(\beta\right)\Arg\alpha}\right|+i\Arg\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)\\ & =\left(\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\alpha\right)+i\Arg\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right) \end{align*} \begin{align*} \beta\Log\alpha & =\left(\Re\left(\beta\right)+i\Im\left(\beta\right)\right)\left(\ln\left|\alpha\right|+i\Arg\left(\alpha\right)\right)\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\left(\alpha\right)+i\left(\Re\left(\beta\right)\Arg\left(\alpha\right)+\Im\left(\beta\right)\ln\left|\alpha\right|\right) \end{align*} となり、\(\Log\alpha^{\beta}=\beta\Log\alpha\)を満たすには、
\[ \Arg\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)=\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\left(\alpha\right) \] となる。
これは任意の\(\alpha\in\mathbb{C}\setminus\left\{ 0\right\} \)について成り立つので、\(\alpha\rightarrow\infty\)とすると、
\[ \lim_{\alpha\rightarrow\infty}\Arg\left(\left(\Im\left(\beta\right)\ln\left|\alpha\right|\right)\right)=\lim_{\alpha\rightarrow\infty}\Im\left(\beta\right)\ln\left|\alpha\right| \] となり、これが成り立つには\(\Im\left(\beta\right)=0\)でないといけない。
そうすると、
\[ \Arg\left(\Re\left(\beta\right)\arg\alpha\right)=\Re\left(\beta\right)\Arg\left(\alpha\right) \] となり、これを満たすには、\(-1<\Re\left(\beta\right)\leq1\)でないといけない。
従って、\(\forall\alpha\in\mathbb{C}\setminus\left\{ 0\right\} ,\Log\alpha^{\beta}=\beta\Log\alpha\Rightarrow-1<\Re\left(\beta\right)\leq1\land\Im\left(\beta\right)=0\Leftrightarrow-1<\beta\leq1\)となる。
故に\(\Rightarrow\)が成り立つ。

\(\Leftarrow\)

条件より、\(-1<\beta\leq1\)なので、\(-1<\Re\left(\beta\right)\leq1\land\Im\left(\beta\right)=0\)であるので、
\begin{align*} \Log\alpha^{\beta} & =\left(\Re\left(\beta\right)\ln\left|\alpha\right|-\Im\left(\beta\right)\Arg\alpha\right)+i\Arg\left(\left(\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\Arg\alpha\right)\right)\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|+i\Arg\left(\Re\left(\beta\right)\Arg\alpha\right)\\ & =\Re\left(\beta\right)\ln\left|\alpha\right|+i\Re\left(\beta\right)\Arg\left(\alpha\right)\\ & =\Re\left(\beta\right)\left(\ln\left|\alpha\right|+i\Arg\left(\alpha\right)\right)\\ & =\Re\left(\beta\right)\Log\left(\alpha\right)\\ & =\beta\Log\left(\alpha\right) \end{align*} となり、\(\Leftarrow\)が成り立つ。

\(\Leftrightarrow\)

これらより\(\Rightarrow\)と\(\Leftarrow\)が成り立つので題意は成り立つ。

(10)

\begin{align*} \Log r^{x} & =\left(\Re\left(x\right)\ln\left|r\right|-\Im\left(x\right)\Arg r\right)+i\Arg\left(\left(\Im\left(x\right)\ln\left|r\right|+\Re\left(x\right)\Arg r\right)\right)\\ & =x\ln\left|r\right|+i\Arg\left(x\Arg r\right)\cmt{\because\Im\left(x\right)=0}\\ & =x\ln\left|r\right|\cmt{\because\Arg r=0} \end{align*}

(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*}