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/ |
| SNSボタン |
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同士の変換
\[
H_{a}\left(x\right)=H_{b}\left(x\right)+\left(a-b\right)\delta_{0,x}
\]
ヘヴィサイドの階段関数とクロネッカーのデルタの関係
\[
H_{a}\left(n\right)-H_{b}\left(n-1\right)=a\delta_{0,n}+\left(1-b\right)\delta_{1,n}
\]
ヘヴィサイドの階段関数の極限表示
\[
H_{\frac{1}{2}}\left(x\right)=\lim_{k\rightarrow\infty}\frac{1}{2}\left(1+\tanh\left(kx\right)\right)
\]