# InverseLaplaceTransform not working

I am trying to find the Inverse Laplace transform of a function I previously obtained from a Laplace transform, but the result obtained does not agree with the initial function. Why is this happening?

LaplaceTransform[2 z^3 Coth[z] Csch[z]^2, z, s] // FullSimplify
InverseLaplaceTransform[%, s, z] // FullSimplify

-(48/(-2 + s)^4) + 3 PolyGamma[1, -1 + s/2] +
1/8 s (12 PolyGamma[2, -1 + s/2] + s PolyGamma[3, -1 + s/2])
-2 E^(2 z) z (3 + z (9 + 4 z) + 3 Coth[z] (1 + 2 z - z Coth[z]))


The last line does not agree with the initial function I was transforming: 2 z^3 Coth[z] Csch[z]^2.

• Have you tried substituting numerical values for z to verify that the expressions are not equivalent? Mar 13, 2020 at 13:30
• Yes, they are definitely different. You can plot those functions and check. Mar 13, 2020 at 14:02
• LaplaceTransform[TrigToExp[2 z^3* Coth[z] *Csch[z]^2] // Simplify, z, s] produces a big expression and InverseLaplace Transform fails with it, returning the input. Mar 13, 2020 at 14:48
• Which version of Mathematica are you using? Mar 13, 2020 at 14:49
• @mikado; Version 12.0 under Windows 10 32-bit on 64-bit machine. Mar 13, 2020 at 15:00

It's seems a BUG in InverseLaplaceTransform.I checked in version 12.0 and 12.1. both occur.

If we try:

InverseLaplaceTransform[ s^2 PolyGamma[3, -1 + s/2], s, z](*Wrong Answer !!!*)
(* 0 *)


We can check and using identity for PolyGamma function:

func = FullSimplify[LaplaceTransform[2*z^3*Coth[z]*Csch[z]^2, z, s]] //
Expand
(*-(48/(-2 + s)^4) + 3 PolyGamma[1, -1 + s/2] +
3/2 s PolyGamma[2, -1 + s/2] + 1/8 s^2 PolyGamma[3, -1 + s/2]*)

func1 = FullSimplify[
InverseLaplaceTransform[ func[[1 ;; 3]], s, z] +
Sum[1/8 InverseLaplaceTransform[s^2*6/(-1 + k + s/2)^4, s,
z], {k, 0, Infinity}], z > 0] // ExpToTrig // Simplify
(* 2 z^3 Coth[z] Csch[z]^2 *)


Result obtained does agree with the initial function.

EDITED:

RE: [CASE:4392492] Bug in Mathematica 12.1 (InverseLaplaceTransform)

It does appear that InverseLaplaceTransform is not behaving correctly/consistently. I have forwarded an issue report to our developers with the information you provided. I also included your contact information in my report.