I am having a slight problem with an integral over a combination of Bessel funcitons and a rational fucntion
Integrate[BesselJ[0, z]^2/(b + z), {z, 0, Infinity}]
where b is a positive constant. The output is
ConditionalExpression[MeijerG[{{1/2, 1/2}, {}}, {{0, 0, 1/2}, {0}}, b^2]/(2*Pi^(3/2)), Im[b] != 0 || Re[b] >= 0]
However, when I do the integral by myself, analyticalally, I get
ConditionalExpression[MeijerG[{{1/2, 1/2}, {}}, {{0, 0, 1/2}, {0}}, b^2]/(4*Pi^(3/2)), Im[b] != 0 || Re[b] >= 0]
which is the same thing multiplied by
1/2
I checked my workings a few times and cannot seem to find a mistake (this doesn't mean that there isn't one - I am not particularly good at contour integration), so I was wondering if there is way of checking the way Mathematica does this calculation. Also it might be a problem of conventions in case Mathematica is using some integral transforms, or any of the functions that are involved - I don't know where to check these for Mathematica, though. Thank you