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International Journal of Engineering Mathematics - Volume 2015 2015, Article ID 523043, 4 pages -

Research Article

Department of Mechanical Engineering, Mississippi State University, P.O. Box 9552, Mississippi State, MS 39762, USA

Cofely UK, Kings Yard, 1 Waterden Road, Queen Elizabeth Olympic Park, London E15 2GP, UK

Received 17 July 2015; Revised 6 October 2015; Accepted 7 October 2015

Academic Editor: Song Cen

Copyright © 2015 Rogelio Luck et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

Autor: Rogelio Luck, Gregory J. Zdaniuk, and Heejin Cho



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