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