Find an Inverse Mellin Transform

Compute an inverse Mellin transform using InverseMellinTransform.

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InverseMellinTransform[Gamma[s], s, x]

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Obtain the strip of holomorphy assumed by InverseMellinTransform.

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InverseMellinTransform[Gamma[s], s, x, GenerateConditions -> True]

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Compute an inverse Mellin transform leading to BesselJ.

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InverseMellinTransform[(2^(-1 + s) a^-s Gamma[1/2 + s/2])/ Gamma[3/2 - s/2], s, x]

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Plot the result for different values of .

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InverseMellinTransform[(2^(-1 + s) a^-s Gamma[1/2 + s/2])/ Gamma[3/2 - s/2], s, x]; Plot[Table[% , {a, 1, 5}] // Evaluate, {x, 0, 7}]

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Create a table of basic inverse Mellin transforms.

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flist = {1/s, 1/(s + 1), a^(s - 1), Gamma[s], Gamma[1 - s], \[Pi] Csc[\[Pi] s], Gamma[s] Sin[(\[Pi] s)/2], Cos[(\[Pi] s)/2] Gamma[s], \[Pi] Cot[\[Pi] s], 1/2 Gamma[s/2]}; TraditionalForm[ Simplify[Grid[ Map[Style[#, ScriptLevel -> 0] &, Join[{{HoldForm@f[s], HoldForm@InverseMellinTransform[f[s], s, x]}}, Transpose[{flist, (InverseMellinTransform[#1, s, x] &) /@ flist}]], {2}], Dividers -> All, Spacings -> {4, 2}, Background -> {None, {{None, GrayLevel[.9]}}, {{1, 1} -> Hue[.6, .4, 1], {1, 2} -> Hue[.6, .4, 1]}}, BaseStyle -> {FontFamily -> Times, FontSize -> 13}]]]

Out[5]//TraditionalForm=