Location of essential singularities of a class of analytic functions

Abstract

A simple method, based on the numerical evaluation of appropriate complex integrals on a closed contour by using appropriate numerical integration rules, is proposed for the location of essential (isolated and non-isolated) singularities of a class of analytic functions in the complex plane. This class of functions consists of those functions where a term in the corresponding Laurent series (in the sum of the negative powers) is missing near the essential singular point and, therefore, it includes odd and even functions among a variety of such functions. The whole approach makes use of the Cauchy residue theorem in elementary complex analysis and requires the values of the analytic function only on a closed contour surrounding the singular point. Numerical results for some elementary transcendental functions are also presented. The present results generalize previous relevant results for zeros and poles of analytic functions and can further be generalized themselves.

Keywords:

• Analytic functions
• Cauchy residue theorem
• complex contour integrals
• essential singularities
• isolated singularities
• non-isolated singularities
• numerical integration
• quadrature rules
• transcendental functions
• trapezoidal quadrature rule

C.R. Categories:

• G.1.m (primary)
• G.1.4 (secondary)