Editing Complex Mexican hat wavelet

In [[applied mathematics]], the '''complex Mexican hat wavelet''' is a low-oscillation, [[complex number|complex-valued]], [[wavelet]] for the [[continuous wavelet transform]]. This wavelet is formulated in terms of its [[Fourier transform]] as the Hilbert [[analytic signal]] of the conventional [[Mexican hat wavelet]]:

:<math>\hat{\Psi}(\omega) = \begin{cases}
  2\sqrt{\frac{2}{3}}\pi^{-\frac{1}{4}}\omega^2 e^{-\frac{1}{2}\omega^2} & \omega\geq0 \\
  0 & \omega\leq 0.
\end{cases}</math>

Temporally, this wavelet can be expressed in terms of the [[error function]],
as:

:<math>\Psi(t) = \frac{2}{\sqrt{3}}\pi^{-\frac{1}{4}}\left(\sqrt{\pi}\left(1 - t^2\right)e^{-\frac{1}{2}t^2} - \left(\sqrt{2}it + \sqrt{\pi}\operatorname{erf}\left[\frac{i}{\sqrt{2}}t\right]\left(1 - t^2\right)e^{-\frac{1}{2}t^2}\right)\right).</math>

This wavelet has <math>O\left(|t|^{-3}\right)</math> [[asymptotic]] temporal decay in <math>|\Psi(t)|</math>,
dominated by the [[Discontinuity (mathematics)|discontinuity]] of the second [[derivative]] of <math>\hat{\Psi}(\omega)</math> at <math>\omega = 0</math>.

This wavelet was proposed in 2002 by Addison ''et al.''<ref>[http://sbe.napier.ac.uk/staff/paddison/wavelet.htm P. S. Addison, ''et al.'', ''The Journal of Sound and Vibration'', 2002] {{webarchive|url=https://archive.today/20000226173323/http://sbe.napier.ac.uk/staff/paddison/wavelet.htm |date=2000-02-26 }}</ref> for applications requiring high temporal precision [[time-frequency analysis]].

== References ==
{{reflist}}

[[Category:Continuous wavelets]]