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Integral with a Bessel function

  • To: mathgroup at
  • Subject: [mg62289] Integral with a Bessel function
  • From: "Alan" <info at>
  • Date: Sat, 19 Nov 2005 05:54:15 -0500 (EST)
  • Sender: owner-wri-mathgroup at

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.



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