符号関数と絶対値

符号関数と絶対値

(1)

\[ \sgn\left(\left|\alpha\right|\beta\right)=\sgn\beta\cnd{\alpha\ne0} \]

(2)

\[ \alpha=\sgn\left(\alpha\right)\left|\alpha\right| \]

(1)

\begin{align*} \sgn\left(\left|\alpha\right|\beta\right) & =\frac{\left|\alpha\right|\beta}{\left|\left|\alpha\right|\beta\right|}\\ & =\frac{\left|\alpha\right|\beta}{\left|\alpha\right|\left|\beta\right|}\\ & =\frac{\beta}{\left|\beta\right|}\\ & =\sgn\beta \end{align*}

(2)

\begin{align*} \alpha & =\begin{cases} \frac{\alpha}{\left|\alpha\right|}\left|\alpha\right| & \alpha\ne0\\ 0 & \alpha=0 \end{cases}\\ & =\begin{cases} \sgn\left(\alpha\right)\left|\alpha\right| & \alpha\ne0\\ 0 & \alpha=0 \end{cases}\\ & =\sgn\left(\alpha\right)\left|\alpha\right| \end{align*}

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符号関数と絶対値

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