Section 8.4 Representations of the Dirac Delta Function
Some other useful representations of the delta function are:
\begin{align*}
\delta(x)
\amp = {1 \over 2\pi}\int_{-\infty}^{\infty} e^{ixt}\, dt\\
\delta(x)
\amp = \lim_{\epsilon\rightarrow 0}\, {1 \over 2\epsilon}
\left[ \Theta(x+\epsilon) - \Theta(x-\epsilon)\right]\\
\delta(x)
\amp = \lim_{\epsilon\rightarrow 0}\,
{1\over \sqrt{2\pi}\, \epsilon}\exp\left(-{x^2 \over 2\epsilon^2}\right)\\
\delta(x)
\amp = {1 \over \pi} \,\lim_{\epsilon\rightarrow 0}\,
{\epsilon \over x^2 + \epsilon^2}\\
\delta(x)
\amp = \lim_{N\rightarrow \infty}\, {\sin Nx \over \pi x}\\
\delta(x)
\amp = {1 \over 2} {d^2 \over dx^2} \vert x \vert\\
\delta(x)
\amp = {1\over \pi^2}\int_{-\infty}^{\infty} {dt\over t(t-x)}
\end{align*}
where Cauchy-Principal Value integration is implied in the last integral. (You can find more limit representations of the delta function at the Wolfram Research Site 1 ).
In quantum mechanics, we sometimes use the closure relation given by:
\begin{equation*}
\delta(x-x')=\sum_{n=0}^\infty \phi_n(x)^*\, \phi_n(x')
\end{equation*}
where the \(\phi_n\) are any complete set of orthonormal eigenfunctions for a hermitian differential operator.
functions.wolfram.com/GeneralizedFunctions/DiracDelta/09/