Finding the zeros of a complex function in PyLab

[PeteyCoco]

I have this characteristic equation for the wave number eigenvalues k_n of a homogeneous infinite cylinder of radius R:

D_{m} = (k_n R) = 0,

where

D_m (z) = n_r J'_m(n_r z)H_m(z) - J_m(n_r z)H'_m(z)

and n_r is the refractive index of the cylinder, the bessel and hankel functions are both of the first kind, and z is a complex argument. I'm not sure how I can solve for the zeros of this. I've been using PyLab, but haven't found any clues as to what I should do.

The article I'm working from is: http://arxiv.org/abs/1302.0245
These are equations (18) and (19) from the article

I'm new to this, so I may be trying something ridiculous.

 

[PeteyCoco]

An even simpler statement of my problem: How can I find the zeros of a complex function in PyLab? Most sources I've found haven't helped.