InverseMellin
InverseMellin[exp, y]
performs the inverse Mellin transform of polynomials in OPE. 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
Overview, DeltaFunction, Integrate2, OPEm, SumS, SumT.
Examples
ym−1
InverseMellin[1/(OPEm + 3), y]
ym+2
ym−1δ(1−y)
InverseMellin[1/OPEm^4, y]
−61ym−1log3(y)
InverseMellin[1/OPEm + 1, y]
ym−1δ(1−y)+ym−1
InverseMellin[1/i + 1, y, i]
yi−1δ(1−y)+yi−1
The inverse operation to InverseMellin
is done by Integrate2
.
Integrate2[InverseMellin[1/OPEm, y], {y, 0, 1}]
m1
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]
m31→21ym−1log2(y)
m21→−ym−1log(y)
ψ(0)(m)→−γym−1δ(1−y)−(1−y1)+ym−1
S1(m−1)→(1−y1)+(−ym−1)
im[SumS[1, OPEm - 1]/(OPEm - 1)]
m−1S1(m−1)→−ym−2log(1−y)
im[SumS[1, OPEm - 1]/(OPEm + 1)]
m+1S1(m−1)→−ym−1+ym−ymlog(1−y)+ymlog(y)
im[SumS[1, -1 + OPEm]/OPEm^2]
m2S1(m−1)→ym−1(ζ(2)−Li2(y)−21log2(y))
im[SumS[1, -1 + OPEm]/OPEm]
mS1(m−1)→ym−1(log(y)−log(1−y))
im[SumS[1, -1 + OPEm]^2/OPEm]
mS12(m−1)→ym−1(−3ζ(2)+Li2(1−y)+2Li2(y)+log2(1−y)+2log2(y))
S2(m−1)→ym−1(ζ(2)δ(1−y)+1−ylog(y))
im[SumS[2, OPEm - 1]/OPEm]
mS2(m−1)→ym−1(ζ(2)−Li2(1−y)−21log2(y))
S3(m−1)→ym−1(ζ(3)δ(1−y)−2(1−y)log2(y))
S11(m−1)→ym−1(1−ylog(1−y))+
S21(m−1)→ym−1(1−yLi2(y)−ζ(2)(1−y1)++2ζ(3)δ(1−y))
im[SumS[1, 1, 1, OPEm - 1]]
S111(m−1)→−21ym−1(1−ylog2(1−y))+