剰余演算の実部と虚部
剰余演算の実部と虚部
\(\left\lfloor z\right\rfloor \)は床関数
\(\left\lceil z\right\rceil \)は天井関数
(1)剰余演算表示
\[ \mod\left(\alpha,\beta\right)=\Re\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)-\Im\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+i\left\{ \Re\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+\Im\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)\right\} \](2)床関数表示
\[ \mod\left(\alpha,\beta\right)=\Re\left(\alpha\right)-\beta\left\lfloor \frac{\Re\left(\alpha\right)\Re\left(\beta\right)+\Im\left(\alpha\right)\Im\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor +i\left(\Im\left(\alpha\right)-\beta\left\lfloor \frac{-\Re\left(\alpha\right)\Im\left(\beta\right)+\Im\left(\alpha\right)\Re\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor \right) \](3)天井関数表示
\[ \mod\left(\alpha,\beta\right)=\Re\left(\alpha\right)+\beta\left\lceil -\frac{\Re\left(\alpha\right)\Re\left(\beta\right)+\Im\left(\alpha\right)\Im\left(\beta\right)}{\left|\beta\right|^{2}}\right\rceil +i\left(\Im\left(\alpha\right)+\beta\left\lceil \frac{\Re\left(\alpha\right)\Im\left(\beta\right)-\Im\left(\alpha\right)\Re\left(\beta\right)}{\left|\beta\right|^{2}}\right\rceil \right) \]-
\(\mod\left(\alpha,\beta\right)\)は剰余演算\(\left\lfloor z\right\rfloor \)は床関数
\(\left\lceil z\right\rceil \)は天井関数
(1)
\begin{align*} \mod\left(\alpha,\beta\right) & =\beta\mod\left(\frac{\alpha}{\beta},1\right)\\ & =\Re\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)-\Im\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+i\left\{ \Re\left(\beta\right)\mod\left(\Im\left(\frac{\alpha}{\beta}\right),1\right)+\Im\left(\beta\right)\mod\left(\Re\left(\frac{\alpha}{\beta}\right),1\right)\right\} \end{align*}(2)
\begin{align*} \mod\left(\alpha,\beta\right) & =\alpha-\beta\left\lfloor \frac{\alpha}{\beta}\right\rfloor \\ & =\alpha-\beta\left\lfloor \frac{\alpha\overline{\beta}}{\left|\beta\right|^{2}}\right\rfloor \\ & =\alpha-\beta\left(\left\lfloor \frac{\Re\left(\alpha\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor +i\left\lfloor \frac{\Im\left(\alpha\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor \right)\\ & =\alpha-\beta\left\lfloor \frac{\Re\left(\alpha\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor -i\beta\left\lfloor \frac{\Im\left(\alpha\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor \\ & =\alpha-\beta\left\lfloor \frac{\Re\left(\alpha\right)\Re\left(\overline{\beta}\right)-\Im\left(\alpha\right)\Im\left(\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor -i\beta\left\lfloor \frac{\Re\left(\alpha\right)\Im\left(\overline{\beta}\right)+\Im\left(\alpha\right)\Re\left(\overline{\beta}\right)}{\left|\beta\right|^{2}}\right\rfloor \\ & =\Re\left(\alpha\right)-\beta\left\lfloor \frac{\Re\left(\alpha\right)\Re\left(\beta\right)+\Im\left(\alpha\right)\Im\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor +i\left(\Im\left(\alpha\right)-\beta\left\lfloor \frac{-\Re\left(\alpha\right)\Im\left(\beta\right)+\Im\left(\alpha\right)\Re\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor \right) \end{align*}(3)
\begin{align*} \mod\left(\alpha,\beta\right) & =\Re\left(\alpha\right)-\beta\left\lfloor \frac{\Re\left(\alpha\right)\Re\left(\beta\right)+\Im\left(\alpha\right)\Im\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor +i\left(\Im\left(\alpha\right)-\beta\left\lfloor \frac{-\Re\left(\alpha\right)\Im\left(\beta\right)+\Im\left(\alpha\right)\Re\left(\beta\right)}{\left|\beta\right|^{2}}\right\rfloor \right)\\ & =\Re\left(\alpha\right)+\beta\left\lceil -\frac{\Re\left(\alpha\right)\Re\left(\beta\right)+\Im\left(\alpha\right)\Im\left(\beta\right)}{\left|\beta\right|^{2}}\right\rceil +i\left(\Im\left(\alpha\right)+\beta\left\lceil \frac{\Re\left(\alpha\right)\Im\left(\beta\right)-\Im\left(\alpha\right)\Re\left(\beta\right)}{\left|\beta\right|^{2}}\right\rceil \right) \end{align*}ページ情報
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剰余演算と床関数・天井関数の関係
\[
\alpha=\beta\left\lfloor \frac{\alpha-\gamma}{\beta}\right\rfloor +\mod\left(\alpha,\beta,\gamma\right)
\]
剰余の剰余
\[
\mod\left(\mod\left(\alpha,n\beta\right),\beta\right)=\mod\left(\alpha,\beta\right)
\]
剰余演算の引数
\[
\mod\left(\alpha,\beta,\gamma\right):=\mod\left(\alpha-\gamma,\beta\right)+\gamma
\]
偏角と剰余の関係
\[
\Arg\alpha=\mod\left(\Arg\left(\alpha\right),-2\pi,\pi\right)
\]