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Problem with an integral that produces a MeijerG function

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

POSTED BY: Bogdan Ganchev
2 Replies

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 BY: Daniel Lichtblau

Thank you very much,

This sounds reasonable - back to pen and paper :)

POSTED BY: Bogdan Ganchev
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