ヘヴィサイドの階段関数の2定義値と関数

ヘヴィサイドの階段関数の2定義値と関数

(1)

\[ f\left(x\right)H\left(\pm1\right)=f\left(\pm x\right)H\left(\pm1\right) \]

(2)

\[ H\left(\pm1\right)=\pm H\left(\pm1\right) \]

(3)

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

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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(x\right)H\left(\pm1\right) & =\begin{cases} f\left(x\right) & \pm1\rightarrow+1\\ 0 & \pm1\rightarrow-1 \end{cases}\\ & =\begin{cases} f\left(\pm x\right) & \pm1\rightarrow+1\\ 0 & \pm1\rightarrow-1 \end{cases}\\ & =f\left(\pm x\right)H\left(\pm1\right) \end{align*}

(2)

\begin{align*} H\left(\pm1\right) & =\left[f\left(x\right)H\left(\pm1\right)\right]_{f\left(x\right)=x\;,\;x=1}\\ & =\left[f\left(\pm x\right)H\left(\pm1\right)\right]_{f\left(x\right)=x\;,\;x=1}\\ & =\pm H\left(\pm1\right) \end{align*}

(2)-2

\begin{align*} H\left(\pm1\right) & =\frac{1\pm1}{2}\\ & =\pm\left(\frac{1\pm1}{2}\right)\\ & =\pm H\left(\pm1\right) \end{align*}

(3)

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

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ヘヴィサイドの階段関数の2定義値と関数

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