expansions in Mathieu functions
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11: 28.15 Expansions for Small
§28.15 Expansions for Small
►§28.15(i) Eigenvalues
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28.15.1
►Higher coefficients can be found by equating powers of
in the following continued-fraction equation, with :
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§28.15(ii) Solutions
…12: Bibliography L
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New method to obtain small parameter power series expansions of Mathieu radial and angular functions.
Math. Comp. 78 (265), pp. 255–274.
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New series expansions for the confluent hypergeometric function
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Appl. Math. Comput. 235, pp. 26–31.
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Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions.
Numer. Math. 116 (2), pp. 269–289.
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature.
Math. Comp. 25 (113), pp. 87–104.
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13: 31.18 Methods of Computation
§31.18 Methods of Computation
►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of ; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.14: Bibliography B
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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V.
Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind.
U.S. Government Printing Office, Washington, D.C..
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A note on Mathieu functions.
Proc. Nederl. Akad. Wetensch. 51 (7), pp. 891–893=Indagationes Math. 10, 319–321 (1948).
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Asymptotic expansions for the coefficient functions that arise in turning-point problems.
Proc. Roy. Soc. London Ser. A 410, pp. 35–60.
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Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case.
In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.),
Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.
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15: Bibliography S
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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions.
J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Uniform asymptotic expansions of modified Mathieu functions.
J. Reine Angew. Math. 247, pp. 1–17.
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Asymptotic expansion of Mellin transforms in the complex plane.
Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels.
SIAM J. Math. Anal. 11 (5), pp. 828–841.
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16: Bibliography D
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On the computation of Mathieu functions.
J. Engrg. Math. 7, pp. 39–61.
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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function.
Methods Appl. Anal. 5 (3), pp. 223–247.
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Uniform asymptotic expansions for prolate spheroidal functions with large parameters.
SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
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Uniform asymptotic approximation of Mathieu functions.
Methods Appl. Anal. 1 (2), pp. 143–168.
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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17: 28.35 Tables
§28.35 Tables
… ►National Bureau of Standards (1967) includes the eigenvalues , for with , and with ; Fourier coefficients for and for , , respectively, and various values of in the interval ; joining factors , for with (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.
§28.35(iii) Zeros
… ►Ince (1932) includes the first zero for , for or , ; 4D. This reference also gives zeros of the first derivatives, together with expansions for small .
Zhang and Jin (1996, pp. 533–535) includes the zeros (in degrees) of , for , and the first 5 zeros of , for or , . Precision is mostly 9S.
18: Bibliography G
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Computational Methods in Special Functions – A Survey.
In Theory and Application of Special Functions (Proc. Advanced
Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis.,
1975), R. A. Askey (Ed.),
pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.
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Computation of the modified Bessel function of the third kind of imaginary orders: Uniform Airy-type asymptotic expansion.
J. Comput. Appl. Math. 153 (1-2), pp. 225–234.
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Mathieu functions.
Trans. Camb. Philos. Soc. 23, pp. 303–336.
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Asymptotische Entwicklungen für unvollständige Gammafunktionen.
Forum Math. 3 (2), pp. 105–141 (German).
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Mathieu functions, a visual approach.
Amer. J. Phys. 71 (3), pp. 233–242.
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19: Bibliography N
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On simplified asymptotic formulas for a class of Mathieu functions.
SIAM J. Math. Anal. 15 (6), pp. 1205–1213.
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On a simplified asymptotic formula for the Mathieu function of the third kind.
SIAM J. Math. Anal. 18 (6), pp. 1616–1629.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors.
2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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Error bounds for the asymptotic expansion of the Hurwitz zeta function.
Proc. A. 473 (2203), pp. 20170363, 16.
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20: Bibliography V
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Accurate calculation of the modified Mathieu functions of integer order.
Quart. Appl. Math. 65 (1), pp. 1–23.
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Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
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Expansion of vacuum magnetic fields in toroidal harmonics.
Comput. Phys. Comm. 81 (1-2), pp. 74–90.
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Symbolic evaluation of coefficients in Airy-type asymptotic expansions.
J. Math. Anal. Appl. 269 (1), pp. 317–331.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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