Software for calculating Bessel functions with an imaginary index

Question

I'm posting this in several places at once to improve response. I've come across an ODE in waveguide problems that can be identified as Bessel's equation with an imaginary index. The formal definition makes sense but I cannot find software that will evaluate this. I've checked boost and matlab. There are a few recent articles about these functions from NIST but they are all theoretical and to not cover implementation.

So, I am reaching out to see if anyone knows of either (1) an open source implementation of these functions, or (2) a transform that allows these functions to be expresses in terms of known functions.

Answer 1

There is the so-called Amos library (written in Fortram), see http://www.netlib.org/amos/ or https://github.com/JuliaLang/openspecfun/tree/master/amos.

See also https://scicomp.stackexchange.com/questions/19902 with a link to a C++ wrapper at https://blog.joey-dumont.ca/complex_bessel/ or https://github.com/valandil/complex_bessel.

Disclaimer I have not tested the libraries.

Answer 2

For some analytic results about these functions: Lebedev has done a lot of work on Bessel $K_{ i\nu}(x)$ functions. These seem to pop up when solving equations in cylindrical coordinates.

Lebedev's book "Special Functions and Their Applications", particularly Chapter 5, has a chance of being of some use (on page 140 I am seeing an asymptotic expansion of $K_{ i\nu}(x)$ for fixed $x>0$ and large $|\nu| \to \infty$).

Maybe it's worth a shot look at some resources involving Kontorovich–Lebedev transforms too; this is an integral transform using $K_{ i\nu}(x)$ as a kernel.