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
\[ f\left(H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)=\left(f\left(0\right)+f\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(f\left(0\right)-f\left(\pm_{1}1\right)\right)H\left(\mp_{2}1\right) \]
\[ f\left(H\left(\pm_{1}1\right)\right)\pm_{2}f\left(-H\left(\mp_{1}1\right)\right)=\left(f\left(0\right)+f\left(\pm_{1}1\right)\right)H\left(\pm_{2}1\right)\mp_{1}\left(f\left(0\right)-f\left(\pm_{1}1\right)\right)H\left(\mp_{2}1\right) \]
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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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| タイトル | 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 |
| URL | https://www.nomuramath.com/rj29dak3/ |
| SNSボタン |
ヘヴィサイドの階段関数の微分・積分と微分・積分表示
\[
\frac{dH\left(x\right)}{dx}=\delta\left(x\right)
\]
ヘヴィサイドの階段関数の複素積分表示
\[
H_{\frac{1}{2}}\left(x\right)=\frac{1}{2\pi i}\lim_{\epsilon\rightarrow0+}\int_{-\infty}^{\infty}\frac{1}{z-i\epsilon}e^{ixz}dz
\]
ヘヴィサイドの階段関数と絶対値・符号関数
\[
H_{a}\left(\left|c\right|x\right)=H_{a}\left(x\right)
\]
ヘヴィサイドの階段関数同士の変換
\[
H_{a}\left(x\right)=H_{b}\left(x\right)+\left(a-b\right)\delta_{0,x}
\]