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expansions in Mathieu functions

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1: 28.11 Expansions in Series of Mathieu Functions
§28.11 Expansions in Series of Mathieu Functions
28.11.7 sin ( 2 m + 2 ) z = n = 0 B 2 m + 2 2 n + 2 ( q ) se 2 n + 2 ( z , q ) .
2: 28.19 Expansions in Series of me ν + 2 n Functions
§28.19 Expansions in Series of me ν + 2 n Functions
3: 28.23 Expansions in Series of Bessel Functions
§28.23 Expansions in Series of Bessel Functions
4: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
5: 28.5 Second Solutions fe n , ge n
For further information on C n ( q ) , S n ( q ) , and expansions of f n ( z , q ) , g n ( z , q ) in Fourier series or in series of ce n , se n functions, see McLachlan (1947, Chapter VII) or Meixner and Schäfke (1954, §2.72). …
6: Bibliography H
  • G. Hunter and M. Kuriyan (1976) Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.
  • 7: 28.34 Methods of Computation
    §28.34(i) Characteristic Exponents
    Methods for computing the eigenvalues a n ( q ) , b n ( q ) , and λ ν ( q ) , defined in §§28.2(v) and 28.12(i), include: …
    §28.34(iv) Modified Mathieu Functions
  • (a)

    Numerical summation of the expansions in series of Bessel functions (28.24.1)–(28.24.13). These series converge quite rapidly for a wide range of values of q and z .

  • (c)

    Use of asymptotic expansions for large z or large q . See §§28.25 and 28.26.

  • 8: 28.2 Definitions and Basic Properties
    §28.2(iv) Floquet Solutions
    §28.2(vi) Eigenfunctions
    9: 28.8 Asymptotic Expansions for Large q
    §28.8 Asymptotic Expansions for Large q
    §28.8(ii) Sips’ Expansions
    §28.8(iii) Goldstein’s Expansions
    Barrett’s Expansions
    Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions me ν ( z , q ) 28.12(ii)) and modified Mathieu functions M ν ( j ) ( z , h ) 28.20(iii)). …
    10: 28.26 Asymptotic Approximations for Large q
    §28.26 Asymptotic Approximations for Large q
    §28.26(i) Goldstein’s Expansions
    The asymptotic expansions of Fs m ( z , h ) and Gs m ( z , h ) in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …
    §28.26(ii) Uniform Approximations
    For asymptotic approximations for M ν ( 3 , 4 ) ( z , h ) see also Naylor (1984, 1987, 1989).