2乗のルート
2乗のルート
\(a\in\mathbb{R}\)とする。
\(a\in\mathbb{R}\)とする。
(1)
\[ \sqrt{\alpha^{2}}=\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \](2)
\[ \sqrt{a^{2}}=\left|a\right| \](3)
\[ \sqrt{-a^{2}}=\left|a\right|i \](4)
\[ \sqrt{\pm a^{2}}=\frac{1}{2}\left(1\pm1+\left(1\mp1\right)i\right)\left|a\right| \](1)
\begin{align*} \sqrt{\alpha^{2}} & =\sqrt{\left|\alpha\right|^{2}\sgn^{2}\left(\alpha\right)}\\ & =\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \end{align*}(1)-2
\begin{align*} \sqrt{\alpha^{2}} & =e^{\frac{1}{2}\Log\left(\alpha^{2}\right)}\\ & =e^{\frac{1}{2}\left(\ln\left|\alpha^{2}\right|+i\Arg\left(\alpha^{2}\right)\right)}\\ & =e^{\ln\left|\alpha\right|+\frac{i}{2}\Arg\left(\left|\alpha\right|^{2}\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|e^{\frac{i}{2}\Arg\left(\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|e^{\frac{1}{2}\Log\left(\sgn\left(\sgn^{2}\left(\alpha\right)\right)\right)}\\ & =\left|\alpha\right|e^{\frac{1}{2}\Log\left(\sgn^{2}\left(\alpha\right)\right)}\\ & =\left|\alpha\right|\sqrt{\sgn^{2}\left(\alpha\right)} \end{align*}(2)
\begin{align*} \sqrt{a^{2}} & =\left|a\right|\sqrt{\sgn^{2}\left(a\right)}\\ & =\left|a\right| \end{align*}(2)-2
\begin{align*} \sqrt{a^{2}} & =e^{\frac{1}{2}\Log\left(a^{2}\right)}\\ & =e^{\frac{1}{2}\ln\left|a^{2}\right|}\\ & =e^{\frac{1}{2}\ln\left|a\right|^{2}}\\ & =e^{\ln\left(\left|a\right|\right)}\\ & =\left|a\right| \end{align*}(3)
\begin{align*} \sqrt{-a^{2}} & =\sqrt{\left(ai\right)^{2}}\\ & =\left|ai\right|\sqrt{\sgn^{2}\left(ai\right)}\\ & =\left|a\right|\sqrt{\sgn^{2}\left(i\right)}\\ & =\left|a\right|i \end{align*}(3)-2
\begin{align*} \sqrt{-a^{2}} & =e^{\frac{1}{2}\Log\left(-a^{2}\right)}\\ & =e^{\frac{1}{2}\left(\ln\left|-a^{2}\right|+i\Arg\left(-a^{2}\right)\right)}\\ & =e^{\frac{1}{2}\left(\ln\left(a^{2}\right)+i\pi\right)}\\ & =ie^{\frac{1}{2}\left(\ln\left(\left|a\right|^{2}\right)\right)}\\ & =ie^{\ln\left(\left|a\right|\right)}\\ & =i\left|a\right| \end{align*}(4)
\begin{align*} \sqrt{\pm a^{2}} & =\sqrt{e^{\frac{\pi i}{2}\left(1\mp1\right)}a^{2}}\\ & =\sqrt{\left(e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right)^{2}}\\ & =\left|e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right|\sqrt{\sgn^{2}\left(e^{\frac{\pi i}{4}\left(1\mp1\right)}a\right)}\\ & =\left|a\right|\sqrt{e^{\frac{\pi i}{2}\left(1\mp1\right)}}\\ & =\left|a\right|e^{\frac{\pi i}{4}\left(1\mp1\right)}\\ & =\left(\frac{1+i}{\sqrt{2}}\right)\left(\frac{1\mp i}{\sqrt{2}}\right)\left|a\right|\\ & =\frac{1}{2}\left(1+\left(1\mp1\right)i\mp\left(-1\right)\right)\left|a\right|\\ & =\frac{1}{2}\left(1\pm1+\left(1\mp1\right)i\right)\left|a\right| \end{align*}(4)-2
\begin{align*} \sqrt{\pm a^{2}} & =e^{\frac{1}{2}\Log\left(\pm a^{2}\right)}\\ & =e^{\frac{1}{2}\left(\Log\left(a^{2}\right)+\Log\left(\pm1\right)\right)}\\ & =e^{\frac{1}{2}\left(\Log\left(a^{2}\right)+\frac{1\mp1}{2}i\pi\right)}\\ & =e^{\frac{i\pi}{4}}e^{\frac{\mp i\pi}{4}}e^{\frac{1}{2}\left(\Log\left(\left|a\right|^{2}\right)\right)}\\ & =e^{\frac{i\pi}{4}\left(1\mp1\right)}\left|a\right|\\ & =\frac{1}{2}\left(1\pm1+\left(1\mp1\right)i\right)\left|a\right| \end{align*}ページ情報
タイトル | 2乗のルート |
URL | https://www.nomuramath.com/xdf7ycjx/ |
SNSボタン |
偏角・対数と符号関数の関係
\[
\Arg\left(z\right)=-i\Log\left(\sgn\left(z\right)\right)
\]
偏角・対数の和と差
\[
\Arg\alpha+\Arg\beta=\Arg\left(\alpha\beta\right)+2\pi\mzp_{-1,0}\left(-\pi,\pi;\Arg\alpha+\Arg\beta\right)
\]
偏角・対数の極限
\[
\lim_{x\rightarrow\pm0}\left\{ \Arg\left(\alpha x\right)-\Arg\left(x\right)\right\} =\begin{cases}
\Arg\alpha & x\rightarrow+0\\
\Arg\left(-\alpha\right)-\pi & x\rightarrow-0
\end{cases}
\]
指数関数の実部と虚部
\[
\left|\alpha^{\beta}\right|=\left|\alpha\right|^{\Re\left(\beta\right)}e^{-\Im\left(\beta\right)\Arg\left(\alpha\right)}
\]