ヘヴィサイドの階段関数の問題

ヘヴィサイドの階段関数の問題

(1)

\[ f\left(H\left(\pm_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)=\left\{ f\left(0\right)g\left(0\right)+f\left(\pm1\right)g\left(\mp1\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(0\right)-f\left(\pm_{1}1\right)g\left(\mp_{1}1\right)\right\} H\left(\mp_{2}1\right) \]

(2)

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right) & =\left\{ f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\mp_{2}1\right) \end{align*}

(3)

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right) & =\left(f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(0\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\mp_{2}1\right) \end{align*}

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\(H\left(x\right)\)はヘヴィサイドの階段関数

(1)

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right) & =\left[h\left(H\left(\pm_{1}1\right)\right)\pm_{2}h\left(-H\left(\mp_{1}1\right)\right)\right]_{h\left(x\right)=f\left(x\right)g\left(-x\right)}\\ & =\left[\left(h\left(0\right)+h\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(h\left(0\right)-h\left(\pm_{1}1\right)\right)H\left(\mp_{2}1\right)\right]_{h\left(x\right)=f\left(x\right)g\left(-x\right)}\\ & =\left\{ f\left(0\right)g\left(0\right)+f\left(\pm1\right)g\left(\mp1\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(0\right)-f\left(\pm_{1}1\right)g\left(\mp_{1}1\right)\right\} H\left(\mp_{2}1\right) \end{align*}

(2)

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right) & =f\left(H\left(\pm_{1}1\right)\right)g\left(\mp_{1}1+H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(\mp_{1}1-H\left(\mp_{1}1\right)\right)\\ & =\left[h\left(H\left(\pm_{1}1\right)\right)\pm_{2}h\left(-H\left(\mp_{1}1\right)\right)\right]_{h\left(x\right)=f\left(x\right)g\left(\mp_{1}1+x\right)}\\ & =\left[\left(h\left(0\right)+h\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(h\left(0\right)-h\left(\pm_{1}1\right)\right)H\left(\mp_{2}1\right)\right]_{h\left(x\right)=f\left(x\right)g\left(\mp_{1}1+x\right)}\\ & =\left\{ f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(\mp_{1}1\pm_{1}1\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(\mp_{1}1\pm_{1}1\right)\right\} H\left(\mp_{2}1\right)\\ & =\left\{ f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\mp_{2}1\right) \end{align*}

(3)

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right) & =f\left(H\left(\pm_{1}1\right)\right)g\left(\mp_{1}1+H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(\mp_{1}1-H\left(\mp_{1}1\right)\right)\\ & =\left[h\left(H\left(\pm_{1}1\right)\right)\pm_{2}h\left(-H\left(\mp_{1}1\right)\right)\right]_{h\left(x\right)=f\left(x\right)g\left(\mp1+x\right)}\\ & =\left[\left(h\left(0\right)+h\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(h\left(0\right)-h\left(\pm_{1}1\right)\right)H\left(\mp_{2}1\right)\right]_{h\left(x\right)=f\left(x\right)g\left(\mp_{1}1+x\right)}\\ & =\left(f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(\mp_{1}1\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(\mp_{1}1\pm_{1}1\right)\right\} H\left(\mp_{2}1\right)\\ & =\left(f\left(0\right)g\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)g\left(0\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(\mp_{1}1\right)-f\left(\pm_{1}1\right)g\left(0\right)\right\} H\left(\mp_{2}1\right) \end{align*}

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ヘヴィサイドの階段関数の問題

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