On the computation and applications of bessel functions with pure imaginary indices (orders) in physics and specifically in corpuscular optics

Bessel functions with pure imaginary index (order) play an important role in corpuscular optics where they govern the dynamics of charged particles in isotrajectory quadrupoles. Much more important role Bessel functions with pure imaginary index play in mathematical physics as solutions to Laplace’s equation by method of variable separation in cylindrical coordinates. Also it was shown recently that such functions appear as solutions to Lame’s equations which are used to characterize the displacement fields in semiconductor nanostructures. But there is no concrete algorithm that computes Bessel functions with pure imaginary order despite the fact that they were discovered long ago. In this paper an algorithm which can be used for the computation of the normal and modified Bessel functions with pure imaginary index is proposed. The developed algorithm is very fast to compute and for small arguments converges after a few iterations.