# Use Meijer-G function to represent elementary functions

I want to represent these elementary functions: $x^{2}\sqrt{x}$, $\sin{4x}$, and $x\ln{x}$ as cases of MeijerG. What arguments should I give to MeijerG to get these elementary functions?

For reference, SymPy is able to represent them. http://docs.sympy.org/0.7.2/modules/integrals/g-functions.html

• Why do you go through the stage of the integration? Why not merely ask (for your first question) how to express $a\ x^{3.5}$ in terms of MeijerG? – David G. Stork Jan 10 '16 at 3:09
• Unfortunately, there's no argument that available in "MeijerG" to represent the integrand in MeijerG form. – làntèrn Jan 10 '16 at 3:18
• I do not know of a direct Mathematica conversion routine for this. Might get some help from e.g. 1, 2, 3. – Daniel Lichtblau Jan 10 '16 at 16:59

## 3 Answers

A partial answer using the reference that you provided:

Limit[b^-1 MeijerG[{{0}, {}}, {{0}, {}}, x^(-5/2)/b], b -> 0]

(*  x^(5/2)  *)

Assuming[{x >= 0},
Sqrt[π] MeijerG[{{}, {}}, {{1/2}, {0}}, 4 x^2] // Simplify]

(*  Sin[4 x]  *)


Version 11 now has the function MeijerGReduce[], which returns equivalent MeijerG[] expressions in an inactive state. Applied to the OP's examples:

MeijerGReduce[{x^2, Sqrt[x], Sin[4 x], x Log[x]}, x]
{x^2, Sqrt[x],
Sqrt[π] Inactive[MeijerG][{{}, {}}, {{1/2}, {0}}, 2 x, 1/2],
x (Inactive[MeijerG][{{1, 1}, {}}, {{1}, {0}}, x] -
Inactive[MeijerG][{{1, 1}, {}}, {{1}, {0}}, x, -1])}


where we see that only the last two functions were touched. The first two fail precisely because their Mellin transforms are not in a form suitable for the conversion:

MellinTransform[{x^2, Sqrt[x]}, x, s]
{DiracDelta[2 + s], 2 DiracDelta[1 + 2 s]}


Contrast this with

MellinTransform[Sin[4 x], x, s]
4^-s Gamma[s] Sin[(π s)/2]


In its current form, it does not seem to fit the style of the defining Mellin-Barnes integral for Meijer $G$, but using the reflection formula and the duplication formula for the gamma function, we have the identity

4^-s Gamma[s] Sin[(π s)/2] ==
Sqrt[π]/2 Gamma[1/2 + s/2]/Gamma[1 - s/2] 2^-s // FullSimplify
True


which can now be compared with the $G$ expression returned by MeijerGReduce[]: Sqrt[π] MeijerG[{{}, {}}, {{1/2}, {0}}, 2 x, 1/2].

There is an undocumented(!) function that essentially performs a Mellin transform through lookup. In your case, however, only one of your four examples actually has a sensible Mellin transform (and thus, a Meijer $G$ representation):

Integrate[Log[x], {x, 0, 1}]; (* force autoloading of the internal function *)

IntegrateImproperDumpMellin[Sin[4 x], x]
(* Sqrt[Pi] IntegrateImproperDumpMeijerGfunction[{}, {}, {1/2}, {0}, 4 x^2] *)


Check:

Simplify[% /. IntegrateImproperDumpMeijerGfunction[a_, b_, c_, d_, z_] :>
MeijerG[{a, b}, {c, d}, z], x > 0]
(* Sin[4 x] *)


Try a more elaborate example:

IntegrateImproperDumpMellin[Hypergeometric2F1[1/2, -1/3, 1, x], x]
(* IntegrateImproperDumpMeijerGfunction[{4/3, 1/2}, {}, {0}, {0}, -x]/(Sqrt[Pi] Gamma[-1/3]) *)


As it is a lookup-based function, it might return unevaluated on functions that nevertheless have a $G$ function representation:

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

MeijerG[{{0, 0}, {}}, {{0}, {-1}}, x]
(* Log[1 + x]/x *)