Re: Bessel integral - Strange Hypergeometric Function
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- Subject: [mg128812] Re: Bessel integral - Strange Hypergeometric Function
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Wed, 28 Nov 2012 03:15:34 -0500 (EST)
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Am 27.11.2012 09:51, schrieb Polal2is:
> Dear all,
> I need to evaluate analytically the integral of a Bessel function J_0(k*r) times an simple rational function 1/(k-k0) (k is the integration variable and k0 a constant) and Mathematica does the job pretty well with a little help.
>
> The point is that I get a function that is very unclear in the output. Indeed, in addition to a J_0 function and a Struve function H_0, I get a:
> "Hypergeometric0F1Regularized^(1,0)(1,-(1/4)k0^2 r^2)"
> which could be very nice if there wasn't this "(1,0)" exponent. Since it would give nothing but another J_0 function.
>
> Usually such exponent refers of course to the derivatives of the function but here it cannot be since: first there is only one variable "r" and second, I checked plotting the derivative together with this function that they are very different. Furthermore the derivative of "Hypergeometric0F1Regularized" is known by Mathematica so it should be explicitely given in that case. Very strange...
>
> I really need to figure out what it is to have a usable analytical solution. The corresponding input is:
>
> Assuming[{k> 0, k0> 0, r> 0}, Integrate[BesselJ[0, k r]/(k - k0), {k, 0,+ [Infinity]}, PrincipalValue -> True]] // TraditionalForm
Its the derivative with respect to the first variable, the index 1.
Derivative with respect to a constant you can make nonsense of by
writing eg.
Limit[ (D[ e^(I pi)),I], I-> 0 ] -> pi
But the integral indices here in these formulas are simply special
values of the general function of two variables.
Nevertheless, derivatives of hypergeometric series with ascending Gamma
factors are nasty.
The formula you need is
(No check of correctness applied. Check yourself analytic against
NIntegrate. Sometimes there are differences in definitions.)
Integrate[BesselJ[0,x]/(x+r),{x,0,oo}] -> Pi/2 (StruveH[r]-BesselY[0,r])
for -Pi < Arg[r] < Pi
For r<0 real there is a rather complicated formula with two
hypergeometric 1_F_2 series.
--
Roland Franzius