MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: kelvin functions ker and kei

  • To: mathgroup at
  • Subject: [mg49141] Re: kelvin functions ker and kei
  • From: Paul Abbott <paul at>
  • Date: Mon, 5 Jul 2004 04:54:20 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <cbrcik$ird$>
  • Sender: owner-wri-mathgroup at

In article <cbrcik$ird$1 at>, mss4 at wrote:

> Does anyone know of a good way to way to integrate ker_2(r), kei_2(r),
> ker_0(r) and kei_0(r) from r=0 to r=Infinity?  

I think you will find it rather difficult to integrate kei_2(r) from r=0 
to r=Infinity since it has a 1/r^2 singularity:

  ke[n_][r_] = Exp[(-1/2) n Pi I] BesselK[n, Exp[(-1/4) Pi I] r]

  Series[ke[2][r], {r, 0, 3}]

> can it be done analytically?  

Yes. After removing the r^(-2) singularity in kei_2(r), all 4 integrals 
are the same and equal to Pi/Sqrt[8].

> I'm also a little unsure about how mathematica does this
> numerically.
> ker_n(r) + I kei_n(r) = Exp[(-1/2)*n*Pi*I] BesselK[n, Exp[(-1/4)*Pi*I]*r]
> any help would be greatly appreciated.

You can compute the indefinite integral for n=0 in closed form and then 
use the asymptotic formulae for BesselK and StruveH at

Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at 

  • Prev by Date: RE: Re: Bug in FromDate
  • Next by Date: Re: Joining 2D arrays
  • Previous by thread: Re: Solid plot
  • Next by thread: Re: Periodic Table of Elements (Chemistry)