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 8h revised Numerical inverse Laplace-Hankel transform added 1 character in body 8h comment Numerical inverse Laplace-Hankel transform Thanks for clarification, now it's clearer. BTW, what's your Mathematica version? I just tested in v9.0.1 and v10.4.1 (Wolfram Cloud), Mathematica is able to correctly find the closed form of Hankel transform of all 3 residues: i.stack.imgur.com/24U1a.png 8h revised Numerical inverse Laplace-Hankel transform Add a remark about a bug in v9 1d revised Numerical inverse Laplace-Hankel transform edited tags 1d comment Numerical inverse Laplace-Hankel transform Oh, finally here comes an answer! (I had almost accepted the reality that my 500 reputation would go with wind...) Then, sorry for my weak basis of complex analysis, but can you elaborate a little on the "The Bromwich contour then can be distorted to yield a solution consisting of the three Residues plus integrals of the discontinuities along the four branch cuts" part? 1d revised Numerical inverse Laplace-Hankel transform 1. add the missing 1/(2 Pi I) coefficient of inverse Laplace transform. 2. Remove the incorrect description of the integral. (What's highly oscillatory is BesselJ[0, ξ] f[p, ξ], not f[p, ξ] ) 2d revised Numerical inverse Laplace-Hankel transform Grammar mistake fixed. 2d awarded Good Question Apr 27 reviewed Close Multiple Integral of equation including dx prime? Apr 27 reviewed Leave Open vector plot with magnitude as legend Apr 27 reviewed Close The Disc and Washer Methods Apr 27 reviewed Leave Open NSolve won't act on very large powers Apr 27 reviewed Close Solve a system of first order ODEs with variable coefficients Apr 27 comment Solving equations but cannot define argument Works fine here: i.stack.imgur.com/fA9np.png I guess you forgot to Clear some of the variables? Apr 27 revised Solving equations but cannot define argument added 3 characters in body Apr 27 comment Solving equations but cannot define argument What's the definition of H? Apr 27 comment Multiply integrand with -1, and the precision changes? @Xavier OK, done. Apr 27 answered Multiply integrand with -1, and the precision changes? Apr 27 comment Numerical inverse Laplace-Hankel transform @Lukas Thanks for the comment, but sadly "SymbolicProcessing" -> 0 seems not to help here... for this integral the only 2 available methods seem to be "ExtrapolatingOscillatory" and "LevinRule". "SymbolicProcessing" isn't an option for "LevinRule", and doesn't improve the speed of "ExtrapolatingOscillatory", what's more, because of a bug mentioned here, one will be unable to adjust the MaxRecursion option if Method -> {"ExtrapolatingOscillatory", "SymbolicProcessing" -> 0} is used. Apr 26 comment KernelObject appears dead problem You still got an answer after these warning? In my case MMA didn't return an answer for several days so I finally quit the kernel.