Identity from Erdelyi et al.

• To: mathgroup at smc.vnet.net
• Subject: [mg107817] [mg107817] Identity from Erdelyi et al.
• From: "Dr. C. S. Jog" <jogc at mecheng.iisc.ernet.in>
• Date: Sat, 27 Feb 2010 03:14:06 -0500 (EST)

Hi:

I am trying to verify one of the identities stated in Erdelyi et al.
Tables of Integral Transforms', Vol. II, 1954, 19.3, pg. 353, No: 19. I
believe this identity is wrong and hence trying to verify it through
Mathematica. Firstly all the subscripts on the left hand side should be
$\nu$ and not zero (although of course $\nu$ can have the value zero).
Secondly, if $\lambda$ in this identity is assumed to be real, then the
left hand side is real while the right hand side is complex, which is the
reason that I believe it is wrong. I believe that on the right hand side
we should have $K_{\nu}$ instead of $H_{\nu}^(2)$, where $K_{\nu}$ is the
modified Bessel function of the second kind (but note that this is just a
belief, and I am trying to verify using Mathematica is this is true).

When I type

In[1]:=Integrate[x*(BesselJ[0,a*x]*BesselY[0,b*x]-BesselJ[0,b*x]*BesselY[0,a*x])/((lam^2+x^2)*(BesselJ[0,b*x]*BesselJ[0,b*x]+BesselY[0,b*x]*BesselY[0,b*x])),{x,0,Infinity}]

after about 1 hour, I get the same output as the input that I have typed
above.

If posssible I would like to evaluate other integrals of the above type
with (lam^4+x^4) in the denominator instead of (lam^2+x^2) etc. Would
appreciate any help in getting Mathematica to evaluate any of these
integrals.

Regards

C. S. Jog

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