ヘヴィサイドの階段関数の正数と負数の和と差
ヘヴィサイドの階段関数の正数と負数の和と差
\(\sgn\left(x\right)\)は符号関数
\(\delta_{ij}\)はクロネッカーのデルタ
(1)
\[ H_{a}\left(x\right)+H_{1-a}\left(-x\right)=1 \](2)
\[ H_{a}\left(x\right)+H_{b}\left(-x\right)=1+\left(a+b-1\right)\delta_{0,x} \](3)
\[ H\left(\pm1\right)+H\left(\mp1\right)=1 \](4)
\[ H_{a}\left(x\right)-H_{a}\left(-x\right)=\sgn\left(x\right) \](5)
\[ H_{a}\left(x\right)-H_{b}\left(-x\right)=\sgn\left(x\right)+\left(a-b\right)\delta_{0,x} \](6)
\[ H\left(\pm1\right)-H\left(\mp1\right)=\pm1 \]-
\(H\left(x\right)\)はヘヴィサイドの階段関数\(\sgn\left(x\right)\)は符号関数
\(\delta_{ij}\)はクロネッカーのデルタ
(1)
\begin{align*} H_{a}\left(x\right)+H_{1-a}\left(-x\right) & =H_{a}\left(x\right)-H_{1-a}\left(x\right)+1+\left(2\left(1-a\right)-1\right)\delta_{0,x}\\ & =\left(a-\left(1-a\right)\right)\delta_{0,x}+1+\left(1-2a\right)\delta_{0,x}\\ & =-\left(1-2a\right)\delta_{0,x}+1+\left(1-2a\right)\delta_{0,x}\\ & =1 \end{align*}(2)
\begin{align*} H_{a}\left(x\right)+H_{b}\left(-x\right) & =H_{a}\left(x\right)+H_{1-a}\left(-x\right)+\left(b-\left(1-a\right)\right)\delta_{0,-x}\\ & =1+\left(a+b-1\right)\delta_{0,x} \end{align*}(3)
\begin{align*} H\left(\pm1\right)+H\left(\mp1\right) & =\frac{1\pm1}{2}+\frac{1\mp1}{2}\\ & =1 \end{align*}(4)
\begin{align*} H_{a}\left(x\right)-H_{a}\left(-x\right) & =H_{a}\left(x\right)-\left(-H_{a}\left(x\right)+1+\left(2a-1\right)\delta_{0,x}\right)\\ & =2H_{a}\left(x\right)-1+\left(1-2a\right)\delta_{0,x}\\ & =2\left(H_{a}\left(x\right)+\left(\frac{1}{2}-a\right)\delta_{0,x}\right)-1\\ & =2H_{\frac{1}{2}}\left(x\right)-1\\ & =2\frac{\sgn\left(x\right)+1}{2}-1\\ & =\sgn\left(x\right) \end{align*}(5)
\begin{align*} H_{a}\left(x\right)-H_{b}\left(-x\right) & =H_{a}\left(x\right)-\left(H_{a}\left(-x\right)+\left(b-a\right)\delta_{0,-x}\right)\\ & =\sgn\left(x\right)+\left(a-b\right)\delta_{0,x} \end{align*}(6)
\begin{align*} H\left(\pm1\right)-H\left(\mp1\right) & =\frac{1\pm1}{2}-\frac{1\mp1}{2}\\ & =\pm1 \end{align*}ページ情報
| タイトル | ヘヴィサイドの階段関数の正数と負数の和と差 |
| URL | https://www.nomuramath.com/w8d823v9/ |
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ヘヴィサイドの階段関数の微分・積分と微分・積分表示
\[
\frac{dH\left(x\right)}{dx}=\delta\left(x\right)
\]
mzp関数の定義と負数の関係
\[
\mzp_{a,b}\left(x_{1},x_{2};-x\right)=-\mzp_{-b,-a}\left(-x_{2},-x_{1};x\right)
\]
ヘヴィサイドの階段関数の問題
\[
f\left(H\left(\pm_{1}1\right)\right)g\left(-H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)g\left(H\left(\mp_{1}1\right)\right)=\left\{ f\left(0\right)g\left(0\right)+f\left(\pm1\right)g\left(\mp1\right)\right\} H\left(\pm_{2}1\right)\mp_{1}\left\{ f\left(0\right)g\left(0\right)-f\left(\pm_{1}1\right)g\left(\mp_{1}1\right)\right\} H\left(\mp_{2}1\right)
\]
ヘヴィサイドの階段関数の２定義値の和と差
\[
H\left(\pm_{1}1\right)\pm_{2}H\left(\pm_{1}1\right)=H\left(\pm_{2}1\right)\pm_{1}H\left(\pm_{2}1\right)
\]