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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剰余演算同士の和・差
\[
\mod\left(x,a,b\right)+\mod\left(y,a,b\right)=\mod\left(x+y,a,b\right)+a\mzp_{0,1}\left(b\sgn\left(a\right),b\sgn\left(a\right)+\left|a\right|;\sgn\left(a\right)\left(\mod\left(x,a,b\right)+\mod\left(y,a,b\right)\right)\right)
\]
剰余演算の引数
\[
\mod\left(\alpha,\beta,\gamma\right):=\mod\left(\alpha-\gamma,\beta\right)+\gamma
\]
負数の剰余演算
\[
\mod\left(-x,a,b\right)=-\mod\left(x+2b,a,b\right)+a\left|\sgn\left\{ \mod\left(x+2b,a,b\right)-b\right\} \right|+2b
\]
剰余演算の定数倍
\[
\frac{1}{\delta}\mod\left(\alpha,\beta,\gamma\right)=\mod\left(\frac{\alpha}{\delta},\frac{\beta}{\delta},\frac{\gamma}{\delta}\right)
\]