複素数と複素共役の実数での剰余演算

複素数と複素共役の実数での剰余演算

\(a\in\mathbb{R}\)とする。

(1)

\[ \mod\left(\alpha,1\right)=\mod\left(\Re\alpha,1\right)+i\mod\left(\Im\alpha,1\right) \]

(2)

\[ \mod\left(\alpha,a\right)=a\left\{ \mod\left(\Re\left(\frac{\alpha}{a}\right),1\right)+i\mod\left(\Im\left(\frac{\alpha}{a}\right),1\right)\right\} \]

(3)

\[ \mod\left(\overline{\alpha},1\right)=\overline{\mod\left(\alpha,1\right)}+i\left|\sgn\mod\left(\Im\alpha,1\right)\right| \]

(4)

\[ \mod\left(\overline{\alpha},a\right)=\overline{\mod\left(\alpha,a\right)}+ia\left|\sgn\mod\left(\Im\alpha,a\right)\right| \]

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\(\mod\left(\alpha,\beta\right)\)は剰余演算

(1)

\begin{align*} \mod\left(\alpha,1\right) & =\alpha-\left\lfloor \Re\left(\alpha\right)\right\rfloor -i\left\lfloor \Im\left(\alpha\right)\right\rfloor \\ & =\Re\alpha-\left\lfloor \Re\left(\alpha\right)\right\rfloor +i\left(\Im\alpha-\left\lfloor \Im\left(\alpha\right)\right\rfloor \right)\\ & =\mod\left(\Re\alpha,1\right)+i\mod\left(\Im\alpha,1\right) \end{align*}

(2)

\begin{align*} \mod\left(\alpha,a\right) & =a\mod\left(\frac{\alpha}{a},1\right)\\ & =a\left\{ \mod\left(\Re\left(\frac{\alpha}{a}\right),1\right)+i\mod\left(\Im\left(\frac{\alpha}{a}\right),1\right)\right\} \end{align*}

(3)

\begin{align*} \mod\left(\overline{\alpha},1\right) & =\mod\left(\Re\overline{\alpha},1\right)+i\mod\left(\Im\overline{\alpha},1\right)\\ & =\mod\left(\Re\alpha,1\right)+i\mod\left(-\Im\alpha,1\right)\\ & =\mod\left(\Re\alpha,1\right)+i\left(-\mod\left(\Im\alpha,1\right)+\left|\sgn\mod\left(\Im\alpha,1\right)\right|\right)\\ & =\overline{\mod\left(\alpha,1\right)}+i\left|\sgn\mod\left(\Im\alpha,1\right)\right| \end{align*}

(4)

\begin{align*} \mod\left(\overline{\alpha},a\right) & =a\mod\left(\frac{\overline{\alpha}}{a},1\right)\\ & =a\left(\overline{\mod\left(\frac{\alpha}{a},1\right)}+i\left|\sgn\mod\left(\Im\left(\frac{\alpha}{a}\right),1\right)\right|\right)\\ & =a\left(\overline{\frac{1}{a}\mod\left(\alpha,a\right)}+i\left|\sgn\mod\left(\Im\alpha,a\right)\right|\right)\\ & =\overline{\mod\left(\alpha,a\right)}+ia\left|\sgn\mod\left(\Im\alpha,a\right)\right| \end{align*}

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複素数と複素共役の実数での剰余演算

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