Bessel function / MeijerG / Integral bug?

• To: mathgroup at smc.vnet.net
• Subject: [mg24048] Bessel function / MeijerG / Integral bug?
• From: Richard Easther <easther at het.brown.edu>
• Date: Wed, 21 Jun 2000 02:20:08 -0400 (EDT)
• Organization: High Energy Theory Group, Physics Dept, Brown U
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

Can anyone provide any pointers here - I am looking at a rather nasty
pair of integrals, namely:

Integrate[
Exp[-tau + I tau] /tau^3  BesselJ[k, a^2 tau^2]^2 , {tau, 0,
Infinity}]
+ Integrate[
Exp[-tau - I tau] /tau^3   BesselJ[k, a^2 tau^2]^2, {tau, 0,
Infinity}]

(The two integrands are obviously complex conjugates, so the overall
expression is real - moreover, the exp(+/- I tau) terms could be
replaced by a cosine, which would make the expression explicitly real
valued).

When I evaluate this expression , I get:

If[a^2 > 0 && Re[k] > 1/2,
(-I/32*MeijerG[{{1 - k}, {1, 1 + k}},
{{-1/2, -1/4, 0, 1/4, 1/2}, {}}, -1/(64*a^4)])/
(Sqrt[2]*Pi^2), Integrate[BesselJ[k, a^2*tau^2]^2/
(E^((1 - I)*tau)*tau^3), {tau, 0, Infinity}]] +
If[a^2 > 0 && Re[k] > 1/2,
(I/32*MeijerG[{{1 - k}, {1, 1 + k}},
{{-1/2, -1/4, 0, 1/4, 1/2}, {}}, -1/(64*a^4)])/
(Sqrt[2]*Pi^2), Integrate[BesselJ[k, a^2*tau^2]^2/
(E^((1 + I)*tau)*tau^3), {tau, 0, Infinity}]]

which is all very well. Then, stipuating that a^2 > 0 && Re[k] > 1/2
(which is true for any values of a and k that I actually care about) I
find that the resulting sum is (according to Mathematica 4) identically
zero, independently of a and k.

Unfortunately, I have several good reasons for believing that it is not
zero - firstly, numerically evaluating the integrals (via NIntegrate)
yields a non-zero answer for arbitrary values of a and k, and plotting
the integrand with (say) a=1 and k=1 yields something that looks very
unlikely to have an integral of zero. Moreover, the physical intuition
derived from the problem which produced these integrals also strongly
suggests a non zero result.

It could be that bugs are buried somewhere in BesselJ and NIntegrate,
but I doubt it - and suspect that Mathematica's implementation of either
the Bessel function integrals used above or the MeijerG function is
buggy :-)

So can anyone a) point me to patch for this bug if it has been seen
before or b) suggest a reference that gives this integral - it does not
seem to be in Gradshteyn and Ryzhik, however.