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New method to obtain small parameter power series expansions of Mathieu radial and angular functions
Author(s):
T.
M.
Larsen;
D.
Erricolo;
P.
L. E.
Uslenghi.
Journal:
Math. Comp.
78
(2009),
255-274.
MSC (2000):
Primary 33E10
Posted:
April 23, 2008
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Abstract |
References |
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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.
References:
-
- [1]
- M. Abramovitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, Inc, New York, 1970.
- [2]
- F. A. Alhargan, Algorithms for the computation of all Mathieu functions of integer order, ACM Trans. Math. Softw. 26 (2001), no. 3, 390-407.
- [3]
- L. Baker, Mathematical function handbook, McGraw-Hill, 1992.
- [4]
- G. Blanch, Numerical aspects of Mathieu eigenvalues, Rend. Circ. Mat. Paler., 2, vol. 15, 1966, pp. 51-97. MR 0229377 (37:4951)
- [5]
- J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, Electromagnetic and acoustic scattering by simple shapes, Hemisphere Publishing Corporation, New York, 1987. MR 1111017 (92f:78014)
- [6]
- S. M. Canta and D. Erricolo, Exact 2D Scattering Analysis of a slot backed by a semielliptical cavity and covered by a multilayer diaphragm, Radio Sci. (2007), submitted.
- [7]
- S. Caorsi, M. Pastorino, and M. Raffetto, Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions, IEEE Trans. Antennas Propagat. 45 (1997), no. 6, 926-935.
- [8]
- D. S. Clemm, Algorithm 352 Characteristic values and associated solutions of Mathieu's differential equation [S22], Commun. ACM 12 (1969), no. 7, 399-407.
- [9]
- -, Remark on algorithm 352 [S22] Characteristic values and associated solutions of Mathieu's differential equation , Commun. ACM 13 (1970), no. 12, 750.
- [10]
- D. Erricolo, M. D. Lockard, C. M. Butler, and P. L. E. Uslenghi, Numerical analysis of penetration, radiation, and scattering for a 2D slotted semielliptical channel filled with isorefractive material, PIER 53 (2005), 69-89.
- [11]
- D. Erricolo, M. D. Lockard, C. M. Butler, and P. L. E. Uslenghi, Currents on conducting surfaces of a semielliptical-channel-backed slotted screen in an isorefractive environment, IEEE Trans. Antennas Propagat. 53 (2005), no. 7, 2350-2356.
- [12]
- D. Erricolo and P. L. E. Uslenghi, Exact radiation and scattering for an elliptic metal cylinder at the interface between isorefractive half-spaces, IEEE Trans. Antennas Propagat. 52 (2004), no. 9, 2214-2225.
- [13]
- -, Penetration, radiation, and scattering for a cavity-backed gap in a corner, IEEE Trans. Antennas Propagat. 53 (2005), no. 8, 2738-2748.
- [14]
- D. Erricolo, P.L.E. Uslenghi, B. Elnour, and F. Mioc, Scattering by a blade on a metallic plane, Electromagnetics 26 (2005), no. 1, 57-71.
- [15]
- Danilo Erricolo, Algorithm 861: Fortran 90 subroutines for computing the expansion coefficients of Mathieu functions using Blanch's algorithm, ACM Transactions on Mathematical Software 32 (2006), no. 4, 622-634. MR 2290450
- [16]
- M. J. Frisch, Remark on algorithm 352 [S22] Characteristic values and associated solutions of Mathieu's differential equation, Commun. ACM 15 (1972), no. 12, 1074.
- [17]
- S. Goldstein, Mathieu Functions, Camb. Phil. Soc. Trans. 23 (1927), 303-336.
- [18]
- T. M. Habashy, J. A. Kong, and W. C. Chew, Resonance and radiation of the elliptic disk microstrip structure, part I: Formulation, IEEE Trans. Antennas Propagat. 35 (1987), no. 8, 877-886.
- [19]
- T. B. Hansen and A. D. Yaghjian, Low-frequency scattering from two-dimensional perfect conductors, IEEE Trans. Antennas Propagat. 40 (1992), no. 11, 1389-1402.
- [20]
- D. B. Hodge, The calculation of the eigenvalues and eigenvectors of Mathieu's equation, Nasa contractor report, The Ohio State University, Columbus, Ohio, 1972.
- [21]
- E. L. Ince, Tables of elliptic cylinder functions, Roy. Soc. Edin. Proc. 52 (1932), 355-423.
- [22]
- G. C. Kokkorakis and J. A. Roumeliotis, Power series expansions for Mathieu functions with small arguments, Math. Comput. 70 (2001), no. 235, 1221-1235. MR 1709153 (2001j:33018)
- [23]
- T. M. Larsen, P. L. E. Uslenghi, and D. Erricolo, Low-frequency behavior of a slotted semielliptical channel, Digest of National Radio Science Meeting (Boulder, CO, USA), Jan 2004, p. 66.
- [24]
- M. M. S. Lira, H. M. de Oliveira, and R. Jd. S. Cintra, Elliptic-cylindrical wavelets: The Mathieu wavelets, IEEE Signal Process. Lett. 11 (2004), no. 1, 52-55.
- [25]
- É. L. Mathieu, Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique, Jour. de Math. Pures et Appliquées (Jour. de Liouville) 13 (1868), no. 137.
- [26]
- N. W. McLachlan, Theory and Application of Mathieu Functions, Dover Publications, New York, 1964. MR 0174808 (30:5001)
- [27]
- J. Meixner and F. W. Schäfke, Mathieusche funktionen und sphäroidfunktionen, Springer, Berlin, 1954. MR 0066500 (16:586g)
- [28]
- P. M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill Book Company, Inc., 1953. MR 0059774 (15:583h)
- [29]
- S. R. Rengarajan and J.E. Lewis, Mathieu Functions of Integraf Orders and Real Arguments, IEEE Trans. Microw. Theory Tech. 28 (1980), no. 3, 276-277.
- [30]
- J. A. Roumeliotis and S.P. Savaidis, Scattering by an infinite circular dielectric cylinder coating eccentrically an elliptic metallic one, IEEE Trans. Antennas Propagat. 44 (1996), no. 5, 757-763.
- [31]
- -, Scattering by an infinite circular dielectric cylinder coating eccentrically an elliptic dielectric cylinder, IEEE Trans. Antennas Propagat. 52 (2004), no. 5, 1180-1185. MR 2075887
- [32]
- S. P. Savaidis and J. A. Roumeliotis, Scattering by an infinite elliptic dielectric cylinder coating eccentrically a circular metallic or dielectric cylinder, IEEE Trans. Microw. Theory Tech. 45 (1997), no. 10, 1792-1800.
- [33]
- J. K. Shaw, W. M. Henry, and W. R. Winfrey, Weakly guiding analysis of elliptical core step index waveguides based on the characteristic numbers of Mathieu's equation, J. Lightwave Technol. 13 (1995), no. 12, 2359-2371.
- [34]
- R. B. Shirts, Algorithm 721 MTIEU1 and MTIEU2: Two subroutines to Compute Eignevalues and solutions to Mathieu's differential equation for noninteger order, ACM Trans. Math. Softw. 19 (1993), no. 3, 391-406.
- [35]
- -, The computation of Eigenvalues and Solutions of Mathieu's differential equation for noninteger order, ACM Trans. Math. Softw. 19 (1993), no. 3, 377-390.
- [36]
- Staff of the computation Laboratory, Tables relating to Mathieu Functions, second edition ed., Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C., 1967, Aug. 1967.
- [37]
- J. A. Stratton, Electromagnetic theory, McGraw-Hill, New York, 1941.
- [38]
- P. L. E. Uslenghi, Exact penetration, radiation and scattering for a slotted semielliptical channel filled with isorefractive material, IEEE Trans. Antennas Propagat. 52 (2004), no. 6, 1473-1480.
- [39]
- M. Valentino and D. Erricolo, Exact two-dimensional scattering from a slot in a ground plane backed by a semielliptical cavity and covered with an isorefractive diaphram, Radio Sci. 42, no. 6, article number RS6S12, 2007 (DOI 10.1029/2006RS003547).
- [40]
- Visual Numerics, Inc., IMSL Fortran subroutines for mathematical applications, 1994.
- [41]
- E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, 1996, reprinted. MR 1424469 (97k:01072)
- [42]
- S. Wolfram, The Mathematica Book, Fifth ed., Wolfram Media, Inc., 2003.
- [43]
- S. Zhang and J.-M. Jin, Computation of special functions, Wiley, New York, 1996. MR 1406797 (97m:65001)
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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:
10.1090/S0025-5718-08-02114-5
PII:
S 0025-5718(08)02114-5
Received by editor(s):
July 26, 2007
Received by editor(s) in revised form:
October 26, 2007
Posted:
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.
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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