Integral with a Bessel function

*To*: mathgroup at smc.vnet.net*Subject*: [mg62289] Integral with a Bessel function*From*: "Alan" <info at optioncity.REMOVETHIS.net>*Date*: Sat, 19 Nov 2005 05:54:15 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Assume "nu" is some real number. Mathematica (5.0) generates Integrate[x^(2 - nu) BesselJ[nu, x], {x,0,Infinity}] = 2^(1-nu) Sqrt[Pi] / Gamma[nu - (1/2)], which seems fine, but is unable to check convergence and supply a condition on "nu". My question: Do you know the (least restrictive) bound on nu for which this is valid? (and a ref. or quick proof?) Since BesselJ(nu, x) ~ Cos(x - phase)/Sqrt(x) for large x, certainly nu > 5/2 is fine. But the Mathematica answer suggests nu > 1/2. However, nu = 1 seems suspiciously too large. So the answer does not seem trivial. Thanks! alan