対数と偏角の性質

対数と偏角の性質

(1)

\[ \arg(\alpha\beta)=\arg(\alpha)+\arg(\beta) \]

(2)

\[ \arg\alpha^{\beta}=\Im\left(\beta\right)\ln\left|\alpha\right|+\Re\left(\beta\right)\arg\alpha+\arg1 \]

(3)

\[ \arg z^{n}=n\arg z+\arg1 \]

(4)

\[ \arg(1/\alpha)=-\arg(\alpha) \]

(5)

\[ \Arg(\alpha\beta)\ne\Arg(\alpha)+\Arg(\beta) \]

(6)

\[ \log\left(\alpha\beta\right)=\log\alpha+\log\beta \]

(7)

\[ \log\alpha^{\beta}=\beta\log\alpha+\log1 \]

(8)

\[ \log z^{n}=n\log z+\log1 \]

(9)

\[ \log\left(1/z\right)=-\log z \]

(10)

\[ \Log\left(\alpha\beta\right)\ne\Log\alpha+\Log\beta \]

(1)

\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\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)で\(\beta=n\in\mathbb{Z}\)とおくと、
\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+\arg0\\ & =-\arg\alpha \end{align*}

(5)

\(\alpha=\beta=-1\)とすると左辺は0、右辺は\(2\pi\)となるので等号は成り立たない。

(6)

\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*}

(7)

\begin{align*} \log\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*}

(8)

(7)で\(\beta=n\in\mathbb{Z}\)とおくと、

\[ \log\alpha^{n}=n\log\alpha+\log1 \]

(8)-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*}

(9)

(8)より、

\begin{align*} \log\left(1/\alpha\right) & =\log\alpha^{-1}\\ & =-\log\alpha+\log1\\ & =-\log\alpha \end{align*}

(10)

\(\alpha=\beta=-1\)とすると左辺は\(0\)、右辺は\(2\pi\)となるので等号は成り立たない。

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対数と偏角の性質

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