Our research group, Dynamica, has openly published a shared object C++ library to compute Bessel functions of complex argument. It is available ready to compile as a tar archive or, for the initiated, as C++ and Fortran source code on Google Code.
The Fortran code comes from D.E. Amos and his 1986 article. The article clearly details the limits, accuracy and speed of the subroutines. Being tested and proven against the test of time, we use them without modification. A C++ wrapper is defined in a header.
More documentation is available on the Google Code repository. To thank you for reading this, here's a pretty picture!
|"Proof" that the algorithms work.|