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11: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
12.9.1 U ( a , z ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π - δ ( < 3 4 π ) ,
12.9.2 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s , | ph z | 1 4 π - δ ( < 1 4 π ) .
12.9.4 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s ± i Γ ( 1 2 - a ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , - 1 4 π + δ ± ph z 3 4 π - δ .
§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
12: 16.22 Asymptotic Expansions
For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
13: 15.12 Asymptotic Approximations
§15.12(iii) Other Large Parameters
14: 28.8 Asymptotic Expansions for Large q
§28.8 Asymptotic Expansions for Large q
Barrett’s Expansions
The approximations apply when the parameters a and q are real and large, and are uniform with respect to various regions in the z -plane. …
Dunster’s Approximations
15: 13.20 Uniform Asymptotic Approximations for Large μ
§13.20(i) Large μ , Fixed κ
16: 28.26 Asymptotic Approximations for Large q
§28.26 Asymptotic Approximations for Large q
§28.26(ii) Uniform Approximations
17: Bibliography U
  • F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.
  • F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.
  • F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.
  • 18: 12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10 Uniform Asymptotic Expansions for Large Parameter
    §12.10(vi) Modifications of Expansions in Elementary Functions
    Modified Expansions
    19: 13.21 Uniform Asymptotic Approximations for Large κ
    §13.21 Uniform Asymptotic Approximations for Large κ
    20: Bibliography L
  • R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.
  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function F 1 with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.
  • J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function F 1 with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
  • J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.