Global Complex Roots and Poles Finding Algorithm in C × R Domain

An algorithm to find the roots and poles of a complex function depending on two arguments (one complex and one real) is proposed. Such problems are common in many fields of science for instance in electromagnetism, acoustics, stability analyses, spectroscopy, optics, and elementary particle physics. The proposed technique belongs to the class of global algorithms, gives a full picture of solutions in a fixed region
$\Omega \subset \mathbb {C}\times \mathbb {R}$
and can be very useful for preliminary analysis of the problem. The roots and poles are represented as curves in this domain. It is an efficient alternative not only to the complex plane zero search algorithms (which require multiple calls for different values of an additional real parameter) but also to tracking algorithms. The developed technique is based on the generalized Cauchy Argument Principle and Delaunay triangulation in three-dimensional space. The usefulness and effectiveness of the method are demonstrated on several examples concerning the analysis of guides (Anti-Resonant Reflecting Acoustic Waveguide, coaxially loaded cylindrical waveguide, graphene transmission line) and a resonant structure (Fabry-Pérot open resonator).