冪乗の性質
冪乗の主値の性質
\(n\in\mathbb{Z}\)とする。
\[ \pv\left(\pv\alpha^{x}\right)^{\gamma}=\pv\alpha^{\gamma x} \]
\(n\in\mathbb{Z}\)とする。
(1)
\[ \pv\left(\alpha\beta\right)^{\gamma}\ne\pv\alpha^{\gamma}\pv\beta^{\gamma} \](2)
\[ \pv\alpha^{\beta}\pv\alpha^{\gamma}=\pv\alpha^{\beta+\gamma} \](3)
\[ \pv\left(\pv\alpha^{\beta}\right)^{\gamma}\ne\pv\alpha^{\beta\gamma} \](4)
\(x\in\mathbb{R}\;,\;\Log\alpha^{x}=x\Log\alpha\)を満たす偏角のとり方と\(x\)とする。\[ \pv\left(\pv\alpha^{x}\right)^{\gamma}=\pv\alpha^{\gamma x} \]
(5)
\[ \pv\left(\pv\alpha^{\beta}\right)^{n}=\pv\alpha^{n\beta} \](6)
\[ \pv\left(\pv\alpha^{-1}\right)^{\gamma}\ne\pv\alpha^{-\gamma} \]-
\(\pv f\left(z\right)\)は関数\(f\left(z\right)\)の主値(1)
\begin{align*} \pv\left(\alpha\beta\right)^{\gamma} & =e^{\gamma\Log\left(\alpha\beta\right)}\\ & \ne e^{\gamma\left(\Log\alpha+\Log\beta\right)}\\ & =e^{\gamma\Log\alpha}e^{\gamma\Log\beta}\\ & =\pv\alpha^{\gamma}\pv\beta^{\gamma} \end{align*}例
\(\alpha=\beta=-1,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(-1\cdot-1\right)^{\frac{1}{2}}=\pv1^{\frac{1}{2}}=\pv1=1\)となるが、右辺は\(\pv\left(-1\right)^{\frac{1}{2}}\pv\left(-1\right)^{\frac{1}{2}}=\pv i\pv i=i^{2}=-1\)となるので、一般的に\(\pv\left(\alpha\beta\right)^{\gamma}\ne\pv\alpha^{\gamma}\pv\beta^{\gamma}\)となる。(2)
\begin{align*} \pv\alpha^{\beta}\pv\alpha^{\gamma} & =e^{\beta\Log\alpha}e^{\gamma\Log\alpha}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha}\\ & =\pv\alpha^{\beta+\gamma} \end{align*}(3)
\begin{align*} \pv\left(\pv\alpha^{\beta}\right)^{\gamma} & =e^{\gamma\Log\alpha^{\beta}}\\ & \ne e^{\gamma\beta\Log\alpha}\\ & =\pv\alpha^{\beta\gamma} \end{align*}例
\(\alpha=-1,\beta=2,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(\pv\left(-1\right)^{2}\right)^{\frac{1}{2}}=\pv\left(1\right)^{\frac{1}{2}}=\pv\left(1\right)=1\)となるが、右辺は\(\pv\left(-1\right)^{2\cdot\frac{1}{2}}=\pv\left(-1\right)=-1\)となるので、一般的に\(\pv\left(\pv\alpha^{\beta}\right)^{\gamma}\ne\pv\alpha^{\beta\gamma}\)となる。(4)
\(\Log\alpha^{x}=x\Log\alpha\)を満たすので、\begin{align*} \pv\left(\pv\alpha^{x}\right)^{\gamma} & =e^{\gamma\Log\alpha^{x}}\\ & =e^{\gamma x\Log\alpha}\\ & =\pv\alpha^{\gamma x} \end{align*}
(5)
\begin{align*} \pv\left(\pv\alpha^{\beta}\right)^{n} & =e^{n\Log\pv\alpha^{\beta}}\\ & =\exists_{1}m\in\mathbb{Z}\;,\;e^{n\left(\beta\Log\alpha+2\pi im\right)}\\ & =e^{n\beta\Log\alpha}\\ & =\pv\alpha^{n\beta} \end{align*}(6)
\begin{align*} \pv\left(\pv\alpha^{-1}\right)^{\gamma} & =e^{\gamma\Log\left(\left|\alpha^{-1}\right|e^{i\Arg\alpha^{-1}}\right)}\\ & =e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{i\Arg\alpha^{-1}}\right)}\\ & =\exists_{1}n\in\mathbb{Z}\;,\;e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{i\left(-\Arg\alpha+n\Arg1\right)}\right)}\\ & =e^{\gamma\left(\ln\left|\alpha\right|^{-1}+i\Arg e^{-i\Arg\alpha}\right)}\\ & \ne e^{-\gamma\left(\ln\left|\alpha\right|+i\Arg\alpha\right)}\\ & =e^{-\gamma\Log\alpha}\\ & =\pv\alpha^{-\gamma} \end{align*}例
\(\alpha=-1,\gamma=\frac{1}{2}\)とすると左辺は\(\pv\left(\pv\left(-1\right)^{-1}\right)^{\frac{1}{2}}=\pv\left(-1\right)^{\frac{1}{2}}=i\)となるが、右辺は\(\pv\left(-1\right)^{-\frac{1}{2}}=\pv e^{-\frac{1}{2}\Log\left(-1\right)}=\pv e^{-\frac{\pi}{2}i}=-i\)となるので、一般的に\(\pv\left(\pv\alpha^{-1}\right)^{\gamma}\ne\pv\alpha^{-\gamma}\)となる。冪乗の多価関数の性質
\(n\in\mathbb{Z}\)とする。
\(n\in\mathbb{Z}\)とする。
(1)
\begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =\pv\alpha^{\gamma}\pv\beta^{\gamma}\mv1^{\gamma}\\ & =\mv\alpha^{\gamma}\mv\beta^{\gamma}\\ & =\mv\alpha^{\gamma}\pv\beta^{\gamma}\\ & =\pv\alpha^{\gamma}\mv\beta^{\gamma} \end{align*}(2)
\begin{align*} \mv\alpha^{\beta}\mv\alpha^{\gamma} & =\pv\alpha^{\beta+\gamma}\mv1^{\beta}\mv1^{\gamma}\\ & =\mv\alpha^{\beta+\gamma}\mv1^{\beta}\\ & =\mv\alpha^{\beta+\gamma}\mv1^{\gamma} \end{align*}(3)
\[ \mv\alpha^{\beta}\mv\alpha^{n}=\mv\alpha^{\beta+n} \](4)
\[ \mv\left(\mv\alpha^{\beta}\right)^{\gamma}=\mv\alpha^{\beta\gamma}\mv1^{\gamma} \](5)
\[ \mv\left(\mv\alpha^{\beta}\right)^{n}=\mv\alpha^{n\beta} \](6)
\[ \mv\left(\mv\alpha^{\frac{1}{n}}\right)^{\beta}=\mv\alpha^{\frac{\beta}{n}} \](7)
\[ \mv\left(\alpha^{\beta}\right)=\pv\left(\alpha^{\beta}\right)\mv1^{\beta} \]-
\(\mv f\left(z\right)\)は関数\(f\left(z\right)\)を多価関数とする(1)
\begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =e^{\gamma\log\left(\alpha\beta\right)}\\ & =e^{\gamma\left(\log\alpha+\log\beta\right)}\\ & =e^{\gamma\left(\Log\alpha+\log1+\Log\beta+\log1\right)}\\ & =e^{\gamma\left(\Log\alpha+\Log\beta+\log1\right)}\\ & =e^{\gamma\Log\alpha}e^{\gamma\Log\beta}e^{\gamma\log1}\\ & =\pv\alpha^{\gamma}\pv\beta^{\gamma}\mv1^{\gamma} \end{align*} \begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =e^{\gamma\log\left(\alpha\beta\right)}\\ & =e^{\gamma\left(\log\alpha+\log\beta\right)}\\ & =e^{\gamma\log\alpha}e^{\gamma\log\beta}\\ & =\mv\alpha^{\gamma}\mv\beta^{\gamma} \end{align*} \begin{align*} \mv\left(\alpha\beta\right)^{\gamma} & =\pv\alpha^{\gamma}\pv\beta^{\gamma}\mv1^{\gamma}\\ & =\pv\alpha^{\gamma}\mv1^{\gamma}\pv\beta^{\gamma}\\ & =\mv\alpha^{\gamma}\pv\beta^{\gamma} \end{align*} 同様に\(\mv\left(\alpha\beta\right)^{\gamma}=\pv\alpha^{\gamma}\mv\beta^{\gamma}\)となる。(2)
\begin{align*} \mv\alpha^{\beta}\mv\alpha^{\gamma} & =e^{\beta\log\alpha}e^{\gamma\log\alpha}\\ & =e^{\beta\left(\Log\alpha+\log1\right)}e^{\gamma\left(\Log\alpha+\log1\right)}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha+\beta\log1+\gamma\log1}\\ & =\pv\alpha^{\beta+\gamma}\mv1^{\beta}\mv1^{\gamma} \end{align*} \begin{align*} \mv\alpha^{\beta}\mv\alpha^{\gamma} & =e^{\beta\log\alpha}e^{\gamma\log\alpha}\\ & =e^{\beta\left(\Log\alpha+\log1\right)}e^{\gamma\left(\Log\alpha+\log1\right)}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha+\beta\log1+\gamma\log1}\\ & =e^{\left(\beta+\gamma\right)\Log\alpha+\beta\log1+\left(\beta+\gamma\right)\log1}\\ & =e^{\left(\beta+\gamma\right)\log\alpha+\beta\log1}\\ & =\mv\alpha^{\beta+\gamma}\mv1^{\beta} \end{align*} 同様に、\(\mv\alpha^{\beta}\mv\alpha^{\gamma}=\mv\alpha^{\beta+\gamma}\mv1^{\gamma}\)となる。(3)
\begin{align*} \mv\alpha^{\beta}\mv\alpha^{n} & =\mv\alpha^{\beta+n}\mv1^{n}\cmt{\text{(2)より}}\\ & =\mv\alpha^{\beta+n} \end{align*}(4)
\begin{align*} \mv\left(\mv\alpha^{\beta}\right)^{\gamma} & =e^{\gamma\log\alpha^{\beta}}\\ & =e^{\gamma\left(\beta\log\alpha+\log1\right)}\\ & =e^{\beta\gamma\log\alpha+\gamma\log1}\\ & =\mv\alpha^{\beta\gamma}\mv1^{\gamma} \end{align*}(5)
\begin{align*} \mv\left(\mv\alpha^{\beta}\right)^{n} & =\mv\alpha^{\beta n}\mv1^{n}\cmt{\text{(4)より}}\\ & =\mv\alpha^{n\beta} \end{align*}(6)
\begin{align*} \mv\left(\mv\alpha^{\frac{1}{n}}\right)^{\beta} & =\mv\alpha^{\frac{\beta}{n}}\mv1^{\beta}\cmt{\text{(4)より}}\\ & =e^{\frac{\beta}{n}\log\alpha+\beta\log1}\\ & =e^{\beta\left(\frac{1}{n}\log\alpha+\log1\right)}\\ & =e^{\beta\left(\frac{1}{n}\log\alpha\right)}\\ & =\mv\alpha^{\frac{\beta}{n}} \end{align*}(7)
\begin{align*} \mv\left(\alpha^{\beta}\right) & =e^{\beta\log\left(\alpha\right)}\\ & =e^{\beta\left(\Log\left(\alpha\right)+\log1\right)}\\ & =e^{\beta\Log\left(\alpha\right)+\beta\log1}\\ & =\pv\left(\alpha^{\beta}\right)\mv1^{\beta} \end{align*}ページ情報
| タイトル | 冪乗の性質 |
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符号関数の偏角・対数
\[
\Log\sgn\alpha=i\Arg\alpha
\]
指数関数の実部と虚部
\[
\left|\alpha^{\beta}\right|=\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\left(\alpha\right)}
\]
偏角・対数の極限
\[
\lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} =\begin{cases}
\Arg\alpha & x\rightarrow+0\\
\Arg\left(-\alpha\right)-\pi & x\rightarrow-0
\end{cases}
\]
複素数と複素共役の和・差
\[
z\pm\overline{z}=2H\left(\pm1\right)\Re z+2iH\left(\mp1\right)\Im z
\]