Chapter 11: Delta Functions

### Representations of the Dirac Delta Function

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_{\epsilon\rightarrow 0}\, {1 \over 2\epsilon} \left[ \Theta(x+\epsilon) - \Theta(x-\epsilon)\right]\\ \noalign{\medskip} \delta(x) &=& \lim_{\epsilon\rightarrow 0}\, {1\over \sqrt{2\pi}\epsilon}\exp\left(-{x^2 \over 2\epsilon^2}\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. (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: $$\delta(x-x')=\sum_{n=0}^\infty \phi_n(x)\, \phi_n(x')$$ where the $\phi_n$ are a complete set of real orthonormal eigenfunctions for a hermitian differential operator.