InverseMellin

Description

InverseMellin[exp, y] performs the inverse Mellin transform of polynomials in OPEm. The inverse transforms are not calculated but a table-lookup is done. WARNING: do not "trust" the results for the inverse Mellin transform involving SumT's; there is an unresolved inconsistency here (related to (-1)^m).

See also:  DeltaFunction, Integrate2, OPEm, SumS, SumT.

Examples

InverseMellin[1/OPEm, y]

InverseMellin[1/(OPEm + 3), y]

InverseMellin[1, y]

InverseMellin[1/OPEm^4, y]

InverseMellin[1/OPEm + 1, y]

InverseMellin[1/i + 1, y, i]

The inverse operation to InverseMellin is done by Integrate2.

Integrate2[InverseMellin[1/OPEm, y], {y, 0, 1}]

Below is a list of all built-in basic inverse Mellin transforms .

\(list = {1, 1\/\(OPEm + n\), 1\/\(\(-OPEm\) + n\), PolyGamma[0, OPEm], 
        SumS[1, \(-1\) + OPEm], SumS[1, \(-1\) + OPEm]\/\(OPEm - 1\), 
        SumS[1, \(-1\) + OPEm]\/\(1 - OPEm\), 
        SumS[1, \(-1\) + OPEm]\/\(OPEm + 1\), 
        SumS[1, \(-1\) + OPEm]\/OPEm\^2, SumS[1, \(-1\) + OPEm]\/OPEm, 
        SumS[1, \(-1\) + OPEm]\^2\/OPEm, SumS[2, \(-1\) + OPEm], 
        SumS[2, \(-1\) + OPEm]\/OPEm, SumS[3, \(-1\) + OPEm], 
        SumS[1, 1, \(-1\) + OPEm], SumS[1, OPEm - 1]\^2, 
        SumS[1, 2, \(-1\) + OPEm], SumS[2, 1, \(-1\) + OPEm], 
        SumS[1, \(-1\) + OPEm]\^3, 
        SumS[1, \(-1\) + OPEm]\ SumS[2, \(-1\) + OPEm], 
        SumS[1, 1, 1, \(-1\) + OPEm]}; \)

im[z_] := z ⟶InverseMellin[z, y]

im[OPEm^(-3)]

im[OPEm^(-2)]

im[PolyGamma[0, OPEm]]

im[SumS[1, OPEm - 1]]

im[SumS[1, OPEm - 1]/(OPEm - 1)]

im[SumS[1, OPEm - 1]/(OPEm + 1)]

im[SumS[1, -1 + OPEm]/OPEm^2]

im[SumS[1, -1 + OPEm]/OPEm]

im[SumS[1, -1 + OPEm]^2/OPEm]

im[SumS[2, OPEm - 1]]

im[SumS[2, OPEm - 1]/OPEm]

im[SumS[3, OPEm - 1]]

im[SumS[1, 1, OPEm - 1]]

im[SumS[1, 2, OPEm - 1]]

im[SumS[2, 1, OPEm - 1]]

im[SumS[1, 1, 1, OPEm - 1]]

Clear[im, list];