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Section 16.9 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 ).
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.
