I was trying to compute the Hilbert Transform (HT) of Exp[-x^4]
using the well-known formula:
f[w_] := InverseFourierTransform[ I Sgn[x] FourierTransform[g[y],
y, x, FourierParameters -> {1, -1}],
x, w, FourierParameters -> {1, -1}]
Oddly, this gave wrong results.
The first direct Fourier transform step was neatly solved:
FourierTransform[Exp[-x^4], x, w, FourierParameters -> {1, -1}]
yielding:
2 Gamma[5/4] HypergeometricPFQ[{}, {1/2, 3/4}, w^4/256] -
1/4 w^2 Gamma[3/4] HypergeometricPFQ[{}, {5/4, 3/2}, w^4/256]
Then the inverse Fourier transform step:
I InverseFourierTransform[ % Sign[w], w, x,
Assumptions -> Element[w, Reals], FourierParameters -> {1, -1}]
gave results with unexpected complex imaginary parts:
(1/(2 Pi Abs[x]))(-1)^(
3/4) E^x^4 x (Gamma[1/4] Gamma[3/4, x^4] -
I Sqrt[2] Pi ((-1 - I) + GammaRegularized[1/4, x^4]))
as testifies the shocking:
% /. x -> 1 // N
-0.261089 - 2.27256 I
Hard as I could try to work this out, I am at a loss what to think. My tentative guess is that there may be a glitch in Inverse Fourier Transform algos. Attached is the notebook. Any idea out there for a workaround?
Notes:
- For
Exp[-x^2] the process outlined above worked and gave the right result (- 2 DawsonF[x]/Sqrt[Pi]), identical to principal value integration of the convolution with 1/x on the real axis.
- For
Exp[-x ^4], the HT was not fount by direct symbolic integration
Attachments:
