The problem of locating the poles and zeros of complex functions in a finite domain of the complex plane, occurs in many scientific disciplines e.g., dispersion relations in plasma physics, the singularity expansion method in electro-magnetic scattering or antenna problems.

The principle of the argument or the winding number is useful in finding the number of zeros of an analytic function in a given contour. A simple extension of this theorem yields relationships involving the locations of these zeros! The resulting equations can be solved very accurately for the zero locations, thus avoiding initial, guess values, which are required by many other techniques. Examples such as a 20th order polynomial, natural frequencies of a thin wire will be discussed.

This method has been extended to the problem of locating the zeros and poles of a complex meromorphic function $M(s)$ in a specified rectangular or square region of the complex plane. It is assumed that $M(s)$ has to be numerically computed. It is interesting to note that the word “meromorphic” is derived from the Greek meros $(\mu \varepsilon \rho \omicron \zeta)$ = fraction and morph $(\mu \omicron \rho \varphi \eta)$ = form, and means “like a fraction.” In keeping with the origin of the word “meromorphic,” the complex function $M(s)$ considered in this paper will be a ratio of two entire functions of the complex variables. The procedure developed here eliminates the usual 2-dimensional search and replaces it with a direct constructive method for determining the poles of $M(s)$ based on an application of Cauchy’s residue theorem. Two examples, i.e., 1) ratios of polynomials and 2) input impedance of a biconical antenna, are numerically illustrated.

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Last updated: 19 Jul 2024