Re: kelvin functions ker and kei

*To*: mathgroup at smc.vnet.net*Subject*: [mg49141] Re: kelvin functions ker and kei*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Mon, 5 Jul 2004 04:54:20 -0400 (EDT)*Organization*: The University of Western Australia*References*: <cbrcik$ird$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cbrcik$ird$1 at smc.vnet.net>, mss4 at duke.edu 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 http://functions.wolfram.com/BesselAiryStruveFunctions Cheers, Paul -- 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 physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul