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Null Function

A null function $\delta^0(x)$ satisfies

\begin{displaymath}
\int^b_a \delta^0(x)\,dx = 0
\end{displaymath} (1)

for all $a, b$, so
\begin{displaymath}
\int_{-\infty}^\infty \vert\delta^0(x)\vert\,dx = 0.
\end{displaymath} (2)

Like a Delta Function, they satisfy
\begin{displaymath}
\delta_0(x) = \cases{
0 & $x \not= 0$\cr
1 & $x = 0$.\cr}
\end{displaymath} (3)

See also Delta Function, Lerch's Theorem




© 1996-9 Eric W. Weisstein
1999-05-25