Some other useful representations of the delta function are: \begin{eqnarray} \delta(x) &=& {1 \over 2\pi}\int_{-\infty}^{\infty} e^{ixt}\, dt\\ \noalign{\medskip} \delta(x) &=& \lim_{a\rightarrow 0}\, {1 \over 2a} \left[ \Theta(x+a) - \Theta(x-a)\right]\\ \noalign{\medskip} \delta(x) &=& \lim_{\epsilon\rightarrow 0}\, {1\over \sqrt{\pi\epsilon}}\exp\left({-x^2 \over \epsilon}\right)\\ \noalign{\medskip} \delta(x) &=& {1 \over \pi} \,\lim_{\epsilon\rightarrow 0}\, {\epsilon \over x^2 + \epsilon^2}\\ \noalign{\medskip} \delta(x) &=& \lim_{N\rightarrow \infty}\, {\sin Nx \over \pi x}\\ \noalign{\medskip} \delta(x) &=& {1 \over 2} {d^2 \over dx^2} \vert x \vert\\ \noalign{\medskip} \delta(x) &=& {1\over \pi^2}\int_{-\infty}^{\infty} {dt\over t(t-x)} \end{eqnarray} where Cauchy-Principal Value integration is implied in the last integral.

In quantum mechanics, we sometimes use the closure relation given by: $$\delta(x-y)=\sum_{n=0}^\infty \phi_n(x)\, \phi_n(y)$$ where the $\phi_n$ are a complete set of real orthonormal eigenfunctions for a hermitian differential operator.