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New method to obtain small parameter power series expansions of Mathieu radial and angular functions


Authors: T. M. Larsen, D. Erricolo and P. L. E. Uslenghi
Journal: Math. Comp. 78 (2009), 255-274
MSC (2000): Primary 33E10
DOI: https://doi.org/10.1090/S0025-5718-08-02114-5
Published electronically: April 23, 2008
MathSciNet review: 2448706
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Abstract | References | Similar Articles | Additional Information

Abstract: Small parameter power series expansions for both radial and angular Mathieu functions are derived. The expansions are valid for all integer orders and apply the Stratton-Morse-Chu normalization. Three new contributions are provided: (1) explicit power series expansions for the radial functions, which are not available in the literature; (2) improved convergence rate of the power series expansions of the radial functions, obtained by representing the radial functions as a series of products of Bessel functions; (3) simpler and more direct derivations for the power series expansion for both the angular and radial functions. A numerical validation is also given.


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Additional Information

T. M. Larsen
Affiliation: Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607
Email: tlarse1@comcast.net

D. Erricolo
Affiliation: Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607
Email: erricolo@ece.uic.edu

P. L. E. Uslenghi
Affiliation: Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607
Email: uslenghi@uic.edu

DOI: https://doi.org/10.1090/S0025-5718-08-02114-5
Received by editor(s): July 26, 2007
Received by editor(s) in revised form: October 26, 2007
Published electronically: April 23, 2008
Additional Notes: This work was supported by the U.S. Department of Defense under MURI grant F49620-01-1-0436 and by a Fellowship from the Aileen S. Andrew Foundation. The authors would like to thank the reviewers for their useful comments.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.