Integration problem

I'm trying to solve this integral including the MeijerG function using Mathematica :

Integrate[
x^((a*b/2) - 1)*
MeijerG[{{}, {}}, {{0}, {}}, (b *x^(l/k))/r^(
a/2)]*(z - x)^((a*b/2) - 1)*
MeijerG[{{}, {}}, {{0}, {}}, (b *(z - x)^(l/k))/r^(a/2)], {x, 0, z},]


But, it keeps giving me the same integral I write as follows:

Integrate[
E^(-b r^(-a/2) x^(l/k) - b r^(-a/2) (-x + z)^(l/k))
x^(-1 + (a b)/2) (-x + z)^(-1 + (a b)/2), {x, 0, z}]


What can i do to get the output of this integration??

• Are you even sure that there's supposed to be a closed form entirely in terms of Meijer $G$? Commented Aug 6, 2016 at 11:48
• With the explicit settings {a = 1, b = 1, k = 1, l = 2, r = 1}, it still returns the Integrate[], which indicates that Mathematica does not know how to solve it (at least in that form). Commented Aug 6, 2016 at 12:54
• Rubi (apmaths.uwo.ca/~arich) cannot do it, either. Commented Aug 6, 2016 at 12:57
• does it mean that the integral has no answer?? Commented Aug 6, 2016 at 13:08
• All it means is that there might indeed be one, but Mathematica is not sufficiently capable to find it. Commented Aug 6, 2016 at 13:11

With

\$Version

(* "10.4.1 for Microsoft Windows (64-bit) (April 11, 2016)" *)


the integral does have a symbolic solution for k == l (and the spurious comma deleted from just before the final bracket).

Integrate[(x^((a*b/2) - 1)*MeijerG[{{}, {}}, {{0}, {}}, (b *x^(l/k))/r^(a/2)]*
(z - x)^((a*b/2) - 1)*MeijerG[{{}, {}}, {{0}, {}}, (b *(z - x)^(l/k))/r^(a/2)]) /.
k -> l, {x, 0, z}]

(* ConditionalExpression[(E^(-b r^(-a/2) z) z^(-1 + a b) Gamma[(a b)/2]^2)/Gamma[a b],
Re[a b] > 0 && Re[z] > 0 && Im[z] == 0] *)


This solution is not as general as that requested but may still be of use.

• I guess it should be noted at this juncture that OP's integral is an autoconvolution integral… Commented Aug 6, 2016 at 19:06
• i copied and pasted your integration but i didn't get your answer, i got the same integration again!!! Commented Aug 7, 2016 at 9:10
• why it keeps giving me the same integration?!! Commented Aug 7, 2016 at 9:21
• What version and OS are you on, @dalia? Commented Aug 7, 2016 at 10:22
• i have Mathematica version 9 Commented Aug 7, 2016 at 10:26