# Complex Mexican hat wavelet

In applied mathematics, the complex Mexican hat wavelet is a low-oscillation, 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:

$\displaystyle{ \hat{\Psi}(\omega) = \begin{cases} 2\sqrt{\frac{2}{3}}\pi^{-\frac{1}{4}}\omega^2 e^{-\frac{1}{2}\omega^2} & \omega\geq0 \\ 0 & \omega\leq 0. \end{cases} }$

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

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

This wavelet has $\displaystyle{ O\left(|t|^{-3}\right) }$ asymptotic temporal decay in $\displaystyle{ |\Psi(t)| }$, dominated by the discontinuity of the second derivative of $\displaystyle{ \hat{\Psi}(\omega) }$ at $\displaystyle{ \omega = 0 }$.

This wavelet was proposed in 2002 by Addison et al.[1] for applications requiring high temporal precision time-frequency analysis.