Hankel Functions: Exploring Real & Imaginary Arguments

[rj_brown]

Hey guys!

Ok so this one's pretty far out there... I'm looking into a piece of work to do with Bessel functions and I'm trying to extend my work using Hankel functions... but there doesn't seem to be a great deal of literature out there so I was wondering if anyone had any experience of them.

I'm trying to look at comparing Hankel functions of real argument with Hankel functions of imaginary argument (or on an ambitious day ever Hankel functions of complex argument!).

I know that for real argument:
H_n^1 (x)=J_n (x) + iY_n (x) and H_n^2 (x)=J_n (x) - iY_n(x)

But is there an equivalence between H_n^1 (ix) and H_n^2 (ix) and Bessel functions?

I'm asking this because my work takes the Bessel equation and the modified Bessel equation. I know that you can transform the Bessel equation into the modified Bessel equation by the transform of x -> ix and that the solutions of Bessel's equation can be transformed in the same way to give the solutions of the modifed Bessel equation.

The solution of Bessel's equation can be written as a linear combination of the first and second Hankel functions and if these are of real argument then surely the solution of the modified Bessel equation will be given as a linear combination of first and second kind Hankel functions of imaginary argument.

I know this is all a bit long and contrived so sorry for going on, really I'm just looking for any info about Hankel functions of imaginary or complex argument and if they have corresponding Bessel functions.

Thanks guys!

(p.s. sorry about the equations, I have no clue about LaTeX)

 

[Astronuc]

See - http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

http://mathworld.wolfram.com/HankelFunctionoftheFirstKind.html

http://mathworld.wolfram.com/HankelFunctionoftheSecondKind.html

Find a copy of Abramowitz, M. and Stegun, I. A. (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972. There are identities of Bessels, modified Bessel and Hankel functions.