天井関数と床関数の和と差
天井関数と床関数の和と差
\(\left\lfloor z\right\rfloor \)は床関数
\(\left\lceil z\right\rceil \)は天井関数
(1)
\[ \left\lceil z\right\rceil +\left\lfloor z\right\rfloor =2z-\mod\left(2z,1\right)-\mzp_{0,1}\left(0,1;2\mod\left(z,1\right)\right)+\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \](2)
\[ \left\lceil z\right\rceil -\left\lfloor z\right\rfloor =\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \]-
\(\delta_{i,j}\)はクロネッカーのデルタ\(\left\lfloor z\right\rfloor \)は床関数
\(\left\lceil z\right\rceil \)は天井関数
(1)
\begin{align*} \left\lceil z\right\rceil +\left\lfloor z\right\rfloor & =z-\mod\left(z,-1\right)+z-\mod\left(z,1\right)\\ & =2z-\left(\mod\left(z,1\right)-\mod\left(-z,1\right)\right)\\ & =2z-\left(\mod\left(\Re z,1\right)-\mod\left(-\Re z,1\right)+i\left(\mod\left(\Im z,1\right)-\mod\left(-\Im z,1\right)\right)\right)\\ & =2z-\left(\mod\left(2\Re z,1\right)+\mzp_{0,1}\left(0,1;2\mod\left(\Re z,1\right)\right)-\left|\sgn\mod\left(\Re z,1\right)\right|+i\left(\mod\left(2\Im z,1\right)+\mzp_{0,1}\left(0,1;2\mod\left(\Im z,1\right)\right)-\left|\sgn\mod\left(\Im z,1\right)\right|\right)\right)\\ & =2z-\mod\left(2z,1\right)-\mzp_{0,1}\left(0,1;2\mod\left(z,1\right)\right)+\left|\sgn\mod\left(\Re z,1\right)\right|+i\left|\sgn\mod\left(\Im z,1\right)\right|\\ & =2z-\mod\left(2z,1\right)-\mzp_{0,1}\left(0,1;2\mod\left(z,1\right)\right)+\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \end{align*}(2)
\begin{align*} \left\lceil z\right\rceil -\left\lfloor z\right\rfloor & =z-\mod\left(z,-1\right)-\left\{ z-\mod\left(z,1\right)\right\} \\ & =-\mod\left(z,-1\right)+\mod\left(z,1\right)\\ & =\mod\left(z,1\right)+\mod\left(-z,1\right)\\ & =\mod\left(\Re z,1\right)+\mod\left(-\Re z,1\right)+i\left\{ \mod\left(\Im z,1\right)+\mod\left(-\Im z,1\right)\right\} \\ & =\left|\sgn\mod\left(\Re z,1\right)\right|+i\left|\sgn\mod\left(\Im z,1\right)\right|\\ & =\sgn\mod\left(\Re z,1\right)+i\sgn\mod\left(\Im z,1\right) \end{align*}(2)-2
実数の場合\begin{align*} \left\lceil x\right\rceil -\left\lfloor x\right\rfloor & =T\left(x\notin\mathbb{Z}\right)\\ & =F\left(x\in\mathbb{Z}\right)\\ & =\left|\sgn\mod\left(x,1\right)\right|\\ & =\sgn\mod\left(x,1\right) \end{align*}
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床関数と天井関数の性質
\[
\left\lceil z\right\rceil =-\left\lfloor -z\right\rfloor
\]
床関数と天井関数の別表記
\[
\left\lfloor z\right\rfloor =z-\mod\left(z,1\right)
\]
整数を含む床関数と天井関数
\[
\left\lfloor \frac{n}{2}\right\rfloor +\left\lceil \frac{n}{2}\right\rceil =n
\]
床関数と天井関数の定義
\[
\left\lfloor x\right\rfloor =\max\left\{ n\in\mathbb{Z};n\leq x\right\}
\]