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Integral with a Bessel function


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


 



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