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Section 11.4 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.