A new method for the computation of the zeros of analytic functions

• Κύριοι συγγραφείς: Ioakimidis, Nikolaos, Anastasselou, Eleni
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2018
• Θέματα: Zeros; Roots; Poles; Residues; Closed-form formulae; Analytical formulae; Analytic functions; Meromorphic functions; Nonlinear equations; Transcendental equations; Cauchy's integral theorem; Cauchy's integral formula; Complex variables; Complex analysis; Complex contour integrals; Numerical integration; Quadrature rules; Neutron moderation; Μηδενικά; Ρίζες; Πόλοι; Ολοκληρωτικά υπόλοιπα; Τύποι κλειστής μορφής; Αναλυτικοί τύποι; Αναλυτικές συναρτήσεις; Μερόμορφες συναρτήσεις; Μη γραμμικές εξισώσεις; Υπερβατικές εξισώσεις; Ολοκληρωτικό θεώρημα του Cauchy; Ολοκληρωτικός τύπος του Cauchy; Μιγαδικές συναρτήσεις; Μιγαδική ανάλυση; Μιγαδικά επικαμπύλια ολοκληρώματα; Αριθμητική ολοκλήρωση; Κανόνες αριθμητικής ολοκληρώσεως; Επιβράδυνση νετρονίων

A new method for the computation of the zeros of analytic functions (or the poles of meromorphic functions) inside or outside a closed contour C in the complex plane is proposed. This method is based on the Cauchy integral formula (in generalized forms) and leads to closed-form formulae for the zeros (or the poles) if they are no more than four. In general, for m zeros (or poles) these can be evaluated as the zeros of a polynomial of degree m. In all cases, complex contour integrals have to be evaluated numerically by using appropriate numerical integration rules. Several practical algorithms for the implementation of the method are proposed and the method of Abd-Elall, Delves and Reid is rederived by two different approaches as one of these algorithms. A numerical application to a transcendental equation appearing in the theory of neutron moderation is also made and numerical results of high accuracy are easily obtained.