Power series expansions for Mathieu functions with small arguments

Kokkorakis, G.; Roumeliotis, J.

doi:10.1090/S0025-5718-00-01227-8

Power series expansions for Mathieu functions with small arguments
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by G. C. Kokkorakis and J. A. Roumeliotis PDF

Math. Comp. 70 (2001), 1221-1235 Request permission

Abstract:

Power series expansions for the even and odd angular Mathieu functions $\operatorname {Se}_m(h,\operatorname {cos}\theta )$ and $\operatorname {So}_m(h,\operatorname {cos}\theta )$, with small argument $h$, are derived for general integer values of $m$. The expansion coefficients that we evaluate are also useful for the calculation of the corresponding radial functions of any kind.

References

J. H. Richmond, Scattering by a conducting elliptic cylinder with dielectric coating, Radio Sci. 23 (1988), 1061–1066.
R. Holland and V. P. Cable, Mathieu functions and their applications to scattering by a coated strip, IEEE Trans. Electromagn. Compat. 34 (1992), 9–16.
T. M. Habashy, J. A. Kong and W. C. Chew, Scalar and vector Mathieu transform pairs, J. Appl. Phys. 60 (1986), 3395–3399.
F. A. Alhargan and S. R. Judah, Frequency response characteristics of multiport planar elliptic patch, IEEE Trans. Microwave Theory Tech. 40 (1992), 1726–1730.
D. W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed., Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1987. MR 899734
J. E. Lewis and G. Deshpande, Models on elliptical cross-section dielectric-tube waveguides, IEE J. Microwaves, Optics and Acoustics 3 (1979), 147–155.
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), 926–935.
N. B. Kakogiannos and J. A. Roumeliotis, Electromagnetic scattering from an infinite elliptic metallic cylinder coated by a circular dielectric one, IEEE Trans. Microwave Theory Tech. 38 (1990), 1660–1666.
J. A. Roumeliotis, H. K. Manthopoulos and V. K. Manthopoulos, Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one, IEEE Trans. Microwave Theory Tech. 41 (1993), 862–869.
J. A. Roumeliotis and S. P. Savaidis, Cutoff frequencies of eccentric circular-elliptic metallic waveguides, IEEE Trans. Microwave Theory Tech. 42 (1994), 2128–2138.
—, Scattering by an infinite circular dielectric cylinder coating eccentrically an elliptic metallic one, IEEE Trans. Antennas Propagat. 44 (1996), 757–763.
S. P. Savaidis and J. A. Roumeliotis, Scattering by an infinite elliptic dielectric cylinder coating eccentrically a circular metallic or dielectric cylinder, IEEE Trans. Microwave Theory Tech. 45 (1997), 1792–1800.
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759
Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York; John Wiley & Sons, Inc., New York, 1984. Reprint of the 1972 edition; Selected Government Publications. MR 757537
R. B. Shirts, The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order, ACM Trans. Math. Software 19 (1993), 377–390.
—, Algorithm 721 MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order, ACM Trans. Math. Software 19 (1993), 391–406.
N. Toyama and K. Shogen, Computation of the value of the even and odd Mathieu functions of order $N$ for a given parameter $S$ and an argument $X$, IEEE Trans. Antennas and Propagation 32 (1984), no. 5, 537–539. MR 748375, DOI 10.1109/TAP.1984.1143362
Delft Numerical Analysis Group, On the computation of Mathieu functions, J. Engrg. Math. 7 (1973), 39–61. MR 367337, DOI 10.1007/BF01535268
S. R. Rengarajan and J. E. Lewis, Mathieu functions of integral orders and real arguments, IEEE Trans. Microwave Theory Tech. 28 (1980), 276–277.
D. S. Clemm, Characteristic values and associated solutions of Mathieu’s differential equation, Comm. Assoc. Comput. Mach. 12 (1969), 399–407.
W. R. Leeb, Algorithm 537: Characteristic values of Mathieu’s differential equation, ACM Trans. Math. Software 5 (1979), 112–117.
Fayez A. Alhargan, A complete method for the computations of Mathieu characteristic numbers of integer orders, SIAM Rev. 38 (1996), no. 2, 239–255. MR 1391228, DOI 10.1137/1038040
Hanan Rubin, Anecdote on power series expansions of Mathieu functions, J. Math. and Phys. 43 (1964), 339–341. MR 170046
Richard Barakat, Agnes Houston, and Elgie Levin, Power series expansions of Mathieu functions with tables of numerical results, J. Math. and Phys. 42 (1963), 200–247. MR 155024
G. C. Kokkorakis and J. A. Roumeliotis, Acoustic eigenfrequencies in concentric spheroidal-spherical cavities, J. Sound. Vibr. 206 (1997), 287–308.