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Re: A Bessel integral

On 9/28/02 11:54 PM, in article an68bt$s7$1 at, "Vladimir
Bondarenko" <vvb at> wrote:

> The expression for W[m_,n_] returned by Mathematica is wrong.
> To prove, just substitute m = n = 0 which is exactly what you had done
> and observe that the output you had had
> W[0,0]=-(2 EulerGamma + Log[4] + 4 PolyGamma[0, 1/2])/(2 Pi)
> = 0.84564
> was incorrect. The correct answer is 1/2.
> Mathematica can handle the numeric integration successfully
> In[1] := NIntegrate[BesselJ[1, x]*BesselJ[0, x], {x, 0, Infinity},
>        Method -> Oscillatory]
>        (* The warnings are skipped *)
> Out[1] = 0.5

You'll find that W[m=1,n=0]=1/2, so Mathematica gets that right. W[0,0]
diverges. Mathematica gets that wrong.

I note that Mathematica yields a result for

Integrate[BesselJ[m, a*x]*BesselJ[n, b*x],  {x, 0, Infinity}]

that appears to agree with formula 6.512(1) of Gradshteyn and Ryshik (4th
ed., 1965), including the condition b<a. (I say appears because I did not
sort out completely difference in notation for the hypergeometric function.)
If you allow a=b, then you must exclude m=n to avoid a divergent integral.

Although successive updates of Mathematica tend to spell out more and more
exceptions, it's perhaps a good idea to interpret broadly the following
warning from on-line help for Integrate:

"For indefinite integrals, Integrate tries to find results that are correct
for almost all values of parameters."

Indeed, it's especially important to investigate special parameter values of
definite integrals because you can't check by differentiation.

Tom Burton

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