# Problem with an integral that produces a MeijerG function

Posted 9 years ago
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 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
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Posted 9 years ago
 It is straightforward to do a sanity check by using an explicit value for b. Below I verify that symbolic result from Integrate for b=2. Integrate[BesselJ[0, z]^2/(2 + z), {z, 0, Infinity}] 1 1 1 MeijerG[{{-, -}, {}}, {{0, 0, -}, {0}}, 4] 2 2 2 (* Out[2]= ------------------------------------------ *) 3/2 2 Pi N[%] (* Out[3]= 0.529248 *) NIntegrate[BesselJ[0, z]^2/(2 + z), {z, 0, Infinity}] (* Out[4]= 0.528948 *) MeijerG[{{1/2, 1/2}, {}}, {{0, 0, 1/2}, {0}}, b^2]/(2*Pi^(3/2))/.b->2. (* Out[5]= 0.529248 *) So the symbolic result when give the value 2 in the input agrees with quadrature of same agrees with the parametrized symbolic result with 2 substituted in afterward.
Posted 9 years ago
 Thank you very much,This sounds reasonable - back to pen and paper :)
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