I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the analytical formula for a Ricker wavelet. Does such a thing exist?
To make the question more explicit, what is the nicest representation of \begin{align} \mathcal{H}[\mathrm{rck}](t):= \frac{2}{\sqrt{3\sigma}\pi^{5/4}}\mathrm{p.v.} \int_{-\infty}^{\infty} \left(1-\frac{\tau^{2}}{\sigma^{2}}\right)\frac{\exp(-\tau^{2}/2\sigma^{2})}{t-\tau} \mathrm{d}\tau \end{align} without use of the symbol "$\mathrm{p.v.}\displaystyle \int_{-\infty}^{\infty}$" and is amenable to computation using standard libraries in C or C++?
Define the Ricker wavelet such that $\left\|\psi\right\|_{2} = 1$: $$ \psi(t):= \frac{2}{\sqrt{3\sigma\sqrt{\pi}}}\left(1 - \frac{t^{2}}{\sigma^{2}} \right)\exp\left(-\frac{t^{2}}{2\sigma^{2}} \right) $$ Then note that $$ \psi(t) = -\sqrt{\frac{4\sigma^{3}}{3\sqrt{\pi}}}\frac{d^{2}}{dt^{2}}\exp\left(-\frac{t^{2}}{2\sigma^{2}} \right) $$ The Hilbert transform is linear and commutes with differentiation, so $$ \mathcal{H}[\psi](t) = -\sqrt{\frac{4\sigma^{3}}{3\sqrt{\pi}}}\frac{d^{2}}{dt^{2}}\mathcal{H}\exp\left(-\frac{t^{2}}{2\sigma^{2}} \right) $$ It also commutes with scalings, so if $M_{1/\sqrt{2}\sigma}f(t):= f(t/\sqrt{2}\sigma)$, then $$ \mathcal{H}[\psi](t) = -\sqrt{\frac{4\sigma^{3}}{3\sqrt{\pi}}}\frac{d^{2}}{dt^{2}}M_{1/\sqrt{2}\sigma}\mathcal{H}\exp\left(-t^{2}\right) $$ But we know that $\mathcal{H}\exp(-t^{2}) = -\exp(-t^{2})\mathrm{erfi}(t)$, where $\mathrm{erfi}(t):= \frac{2}{\sqrt{\pi}}\int_{0}^{t} \exp(z^{2}) \, \mathrm{d}z$, so $$ \mathcal{H}[\psi](t) = \sqrt{\frac{4\sigma^{3}}{3\sqrt{\pi}}}\frac{d^{2}}{dt^{2}}M_{1/\sqrt{2}\sigma}\exp(-t^{2})\mathrm{erfi}(t) $$ If $\gamma := t/\sqrt{2}\sigma$, then $$ \mathcal{H}[\psi](t) = \frac{1}{\sqrt{3\sigma\sqrt{\pi}}}\frac{d^{2}}{d\gamma^{2}}\exp(-\gamma^{2})\mathrm{erfi}(\gamma) $$ which yields $$ \mathcal{H}[\psi](t) = \frac{1}{\sqrt{3\sigma\sqrt{\pi}}}\left[2e^{-\gamma^{2}}(2\gamma^2-1)\mathrm{erfi}(\gamma) - \frac{4\gamma}{\sqrt{\pi}}\right] $$ or $$ \mathcal{H}[\psi](t)= \frac{2}{\sqrt{3\sigma\sqrt{\pi}}}\left[e^{-t^{2}/2\sigma^2}(t^{2}/\sigma^2-1)\mathrm{erfi}(t/\sqrt{2}\sigma) - \sqrt{\frac{2}{\pi}}\frac{t}{\sigma}\right] $$ Use at your own risk.