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11: 18.32 OP’s with Respect to Freud Weights
However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). … For asymptotic approximations to OP’s that correspond to Freud weights with more general functions Q ( x ) see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).
12: 28.26 Asymptotic Approximations for Large q
§28.26 Asymptotic Approximations for Large q
§28.26(ii) Uniform Approximations
For asymptotic approximations for M ν ( 3 , 4 ) ( z , h ) see also Naylor (1984, 1987, 1989).
13: 2.1 Definitions and Elementary Properties
means that for each n , the difference between f ( x ) and the n th partial sum on the right-hand side is O ( ( x - c ) n ) as x c in X . … Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions. …For an example see (2.8.15). …
§2.1(iv) Uniform Asymptotic Expansions
§2.1(v) Generalized Asymptotic Expansions
14: 15.12 Asymptotic Approximations
§15.12 Asymptotic Approximations
§15.12(i) Large Variable
§15.12(ii) Large c
As λ , … For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).
15: 10.70 Zeros
§10.70 Zeros
Asymptotic approximations for large zeros are as follows. …
16: 13.22 Zeros
Asymptotic approximations to the zeros when the parameters κ and/or μ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …
17: 33.21 Asymptotic Approximations for Large | r |
§33.21 Asymptotic Approximations for Large | r |
§33.21(i) Limiting Forms
  • (b)

    When r ± with ϵ < 0 , Equations (33.16.10)–(33.16.13) are combined with

    33.21.1
    ζ ( ν , r ) e - r / ν ( 2 r / ν ) ν ,
    ξ ( ν , r ) e r / ν ( 2 r / ν ) - ν , r ,
    33.21.2
    ζ ( - ν , r ) e r / ν ( - 2 r / ν ) - ν ,
    ξ ( - ν , r ) e - r / ν ( - 2 r / ν ) ν , r - .

    Corresponding approximations for s ( ϵ , ; r ) and c ( ϵ , ; r ) as r can be obtained via (33.16.17), and as r - via (33.16.18).

  • §33.21(ii) Asymptotic Expansions
    18: 35.9 Applications
    The asymptotic approximations of §35.7(iv) are applied in numerous statistical contexts in Butler and Wood (2002). In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …
    19: Roderick S. C. Wong
    He is the author of the book Asymptotic Approximations of Integrals, published by Academic Press in 1989 and reprinted by SIAM in its Classics in Applied Mathematics Series in 2001, and of Lecture Notes on Applied Analysis, published by World Scientific in 2010. …
  • 20: 36.11 Leading-Order Asymptotics
    §36.11 Leading-Order Asymptotics