# InverseMellinTransform producing two different results for the same input?

Consider the expression

expr = Gamma[1 + s]/Gamma[1 - s] Gamma[-s]^2;


and a slightly simplified version of the same

expr2 = expr//FullSimplify


If we take the InverseMellinTransform of the first shape of the expression expr, we get

InverseMellinTransform[expr, s, x]


Log[1 + x]

On the other hand, if we take the InverseMellinTransform of the simplified form expr2, we get

InverseMellinTransform[expr2, s, x]


Log[1 + 1/x]

Evidently, the two results do not agree. How to make sense of this? I assume, this is not correct behavior? Is there some way to do the transform more carefully in Mathematica, such that both calculations produce the same result consistently?

• at least Mathematica is consistent. It gives back the same result when transforming back :) screen shot !Mathematica graphics may be there is some deep mathematical reason for this. Apr 28 '20 at 23:44

expr = Gamma[1 + s]/Gamma[1 - s] Gamma[-s]^2;

imt1 = InverseMellinTransform[expr, s, x, GenerateConditions -> True]


MellinTransform[imt1 // Normal, x, s, GenerateConditions -> True]


Simplifying expr

expr2 = expr // FullSimplify

(* (π Csc[π s])/s *)

imt2 = InverseMellinTransform[expr2, s, x, GenerateConditions -> True]


MellinTransform[imt2 // Normal, x, s, GenerateConditions -> True]


The inverse transforms are for different regions.

InverseMellinTransform[expr, s, x,
Assumptions -> 0 < Re[s] < 1] //
FullSimplify[#, 0 < x < 1] &

(* Log[1 + 1/x] *)