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)
\[ f\left(\pm_{1}H\left(\pm_{2}1\right)\right)=f\left(0\right)H\left(\mp_{2}1\right)+f\left(\pm_{1}1\right)H\left(\pm_{2}1\right) \]-
\(H\left(x\right)\)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*}ページ情報
| タイトル | ヘヴィサイドの階段関数の２定義値と関数 |
| URL | https://www.nomuramath.com/h01ij6zb/ |
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ヘヴィサイドの階段関数の２定義値の和と差
\[
H\left(\pm_{1}1\right)\pm_{2}H\left(\pm_{1}1\right)=H\left(\pm_{2}1\right)\pm_{1}H\left(\pm_{2}1\right)
\]
ヘヴィサイドの階段関数の極限表示
\[
H_{\frac{1}{2}}\left(x\right)=\lim_{k\rightarrow\infty}\frac{1}{2}\left(1+\tanh\left(kx\right)\right)
\]
ヘヴィサイドの階段関数と単位ステップ関数の定義
\[
H_{a}\left(x\right)=\begin{cases}
0 & \left(x<0\right)\\
a & \left(x=0\right)\\
1 & \left(0<x\right)
\end{cases}
\]
ヘヴィサイドの階段関数と符号関数・絶対値
\[
H_{\frac{1}{2}}\left(\pm x\right)=\frac{1\pm\sgn x}{2}
\]