Re: Complex bessel function

• To: mathgroup at smc.vnet.net
• Subject: [mg75210] Re: Complex bessel function
• From: jesse.woodroffe at gmail.com
• Date: Fri, 20 Apr 2007 00:44:08 -0400 (EDT)
• References: <f079np\$46c\$1@smc.vnet.net>

```Zach,

>From my understanding of Bessel functions, you're not ikely to find
any simple identity to separate the real and imaginary parts. They're
certainly not distributive, nor do they have any real tendency towards
simplicity (elegance, yes; simplicity, no.). Being that there are no
obviously useful relations in Abramowitz and Stegun[1], I'm fairly
certain that learning to love complex arguments mightn't be such a bad
idea.

I will mention in passing, however, that that there is a fairly well
known identity for BesselJ that is an addition theorem of sorts [2]:

BesselJ[n,u+v]=Sum[BesselJ[n -/+ k,u]*BesselJ[k,v],{k,-
Infinity,Infinity}]

This formula is expanded to include BesselY and Hankel functions in
A&S[3].

However, I believe that you'll find that this formula also works for
BesselI with the caveat that v is signed, i.e.

BesselI[n,u+I*v]=Sum[BesselI[n-k,u]*BesselI[k,I*v],{k,-
Infinity,Infinity}]
BesselI[n,u-I*v]=Sum[BesselI[n-k,u]*BesselI[k,-I*v],{k,-
Infinity,Infinity}]

In practice, these summations require far fewer than an infinite
number of terms in order to converge, but in any event they're not
going to separate out your real and complex parts.

Best of luck.

Jesse Woodroffe

References:

[1] http://www.math.sfu.ca/~cbm/aands/page_374.htm
[2] http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
[3] http://www.math.sfu.ca/~cbm/aands/page_363.htm

On Apr 19, 3:37 am, "Grasley, Zachary" <zgras... at civil.tamu.edu>
wrote:
> I am currently trying to isolate the real and imaginary components of
> first and second order modified Bessel functions.  For example, I have a
> function expressed as
>
> F=BesselI[0,sqrt(i)*a+b], where i is the imaginary unit.  Both a and b
> are positive real.  I need to separate out the real and imaginary
> components of F symbolically.  I have tried Re, Im, ComplexExpand, I
> have used Assuming to define a and b as real and 0<a<infinity and
> 0<b<infinity.  I have tried ComplexExpand.  I have tried Refine[Im[F]].
> I loaded the package ReIm.
>
> All of these functions do a fine job separating the real and imaginary
> components when I provide numeric values for a and b, but are unable to
> solve symbolically.  I thought Refine was supposed to solve symbolically
> as if a numeric value has been assigned to the variables (with the value
> range of the variable defined by Assuming or the assumptions listed in
> Refine)?  Am I missing some simple trick to solve this symbolically, or
> is this not possible?
>