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Re: trouble integrating Bessel Functions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg6500] Re: [mg6484] trouble integrating Bessel Functions
*From*: jpk at max.mpae.gwdg.de
*Date*: Thu, 27 Mar 1997 02:42:36 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
> From pankratz at eta.pha.jhu.edu Tue Mar 25 05:50:30 1997
> Date: Mon, 24 Mar 1997 21:38:31 -0500 (EST)
> From: Chris Pankratz <pankratz at eta.pha.jhu.edu>
> To: mathgroup at smc.vnet.net
> Subject: [mg6484] trouble integrating Bessel Functions
>
> I have encountered a curious problem calculating the fourier transform of a
> modified bessel function. Mathematica 2.2 will calculate it just fine, but
> Mathematica 3.0 won't. Can anyone assist? Thanks.
> Chris
>
> Mathematica 2.2 does it just fine:
> ----------------------------------
>
> (1/Sqrt[2 Pi]) * Integrate[BesselK[0, t] Exp[I w t], {t, -Infinity,
> Infinity}]
>
> Pi
> Sqrt[--]
> 2
> ------------
> 2
> Sqrt[1 + w ]
>
>
> Mathematica 3.0 doesn't seem to have the right integration package loaded
> -------------------------------------------------------------------------
> - it just echoes the command back to me, without giving a result.
>
>
Oh, Your first integral is Infinity and Mma 2.2 doesn't report this. Because
You don't want a Besselfunction wit a negative argument. The imaginary part
of K[0,t] diverges for t<0 !
When You give
Integrate[BesselK[0,t]*Exp[I*w*t],{t,0,Infinity},
Assumptions->Im[w]==0]/Sqrt[2 Pi]
You get the correct result.
Hope that helps
Jens
My Abramowitz/Stegun say
Integrate[ t^mu K[mu,t],{t,0,Infinity}==2
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