ヘヴィサイドの階段関数の2定義値を引数に持つ関数の和と差

ヘヴィサイドの階段関数の2定義値を引数に持つ関数の和と差

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

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\(H\left(x\right)\)はヘヴィサイドの階段関数

\begin{align*} f\left(H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right) & =f\left(0\right)H\left(\mp_{1}1\right)+f\left(1\right)H\left(\pm_{1}1\right)\pm_{2}\left(f\left(0\right)H\left(\pm_{1}1\right)+f\left(-1\right)H\left(\mp_{1}1\right)\right)\\ & =f\left(0\right)H\left(\mp_{1}1\right)+f\left(\pm_{1}1\right)H\left(\pm_{1}1\right)\pm_{2}\left(f\left(0\right)H\left(\pm_{1}1\right)+f\left(\pm_{1}1\right)H\left(\mp_{1}1\right)\right)\\ & =f\left(0\right)\left(H\left(\mp_{1}1\right)\pm_{2}H\left(\pm_{1}1\right)\right)+f\left(\pm_{1}1\right)\left(H\left(\pm_{1}1\right)\pm_{2}H\left(\mp_{1}1\right)\right)\\ & =f\left(0\right)\left(H\left(\pm_{2}1\right)\mp_{1}H\left(\mp_{2}1\right)\right)+f\left(\pm_{1}1\right)\left(H\left(\pm_{2}1\right)\pm_{1}H\left(\mp_{2}1\right)\right)\\ & =\left(f\left(0\right)+f\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(f\left(0\right)-f\left(\pm_{1}1\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2定義値を引数に持つ関数の和と差

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