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