Computing the Fourier Transform of $\exp(-\log(x)^2)$

Asked Jan 18 '22 at 12:45

Active Jan 19 '22 at 08:30

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I would like to use the approach of R. E. Fredericksen and R. F. Hess, “Estimating multiple temporal mechanisms in human vision,” Vision Res., vol. 38, no. 7, pp. 1023–1040, Apr. 1998, doi:10.1016/S0042-6989(97)00239-3

The basic function is under the form $\exp(-((\log(x) -log(\tau))/\sigma)^2), x \geq 0, \tau > 0$ which is a pulse centered at $\tau$, the width being controlled by $\sigma$, and asymetric left and right slopes.

I need its Fourier transform. I used various approaches, f.i. https://charlesfrye.github.io/stats/2017/11/22/gaussian-diff-eq.html and How to calculate the Fourier transform of a Gaussian function?. But basically I can't get back to some known, integrable expression. Any idea if such function possess an analytical Fourier transform ?

Regards

Pascal Dupuis

edited Jan 19 '22 at 08:30

asked Jan 18 '22 at 12:45

CdeMills

$\log(x-\tau)$ is not real valued for $x \le \tau$. Do you really intend $ x \ge 0$? – Andy Walls Jan 19 '22 at 00:33
My bad. The original formulation is $-((log(x/\tau)/\sigma)^2)$. Corrected the question. Thanks for pointing it. – CdeMills Jan 19 '22 at 08:26

Computing the Fourier Transform of $\exp(-\log(x)^2)$

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