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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:
Temporally, this wavelet can be expressed in terms of the error function,as:
This wavelet has O {\displaystyle O\left} asymptotic temporal decay in | Ψ | {\displaystyle |\Psi |} ,dominated by the discontinuity of the second derivative of Ψ ^ {\displaystyle {\hat {\Psi }}} at ω = 0 {\displaystyle \omega =0}.
This wavelet was proposed in 2002 by Addison et al. for applications requiring high temporal precision time-frequency analysis.