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1: 31.13 Asymptotic Approximations
§31.13 Asymptotic Approximations
For asymptotic approximations for the accessory parameter eigenvalues q m , see Fedoryuk (1991) and Slavyanov (1996). For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
2: 16.22 Asymptotic Expansions
§16.22 Asymptotic Expansions
Asymptotic expansions of G p , q m , n ( z ; a ; b ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
3: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
The uniform asymptotic approximations given in §14.15 for P ν - μ ( x ) and Q ν μ ( x ) for 1 < x < are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
4: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. …
5: 2.2 Transcendental Equations
§2.2 Transcendental Equations
2.2.1 f ( x ) x , x .
An important case is the reversion of asymptotic expansions for zeros of special functions. …where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …For other examples see de Bruijn (1961, Chapter 2).
6: 32.12 Asymptotic Approximations for Complex Variables
§32.12 Asymptotic Approximations for Complex Variables
7: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as n , with x and other parameters fixed, for continuous q -ultraspherical, big and little q -Jacobi, and Askey–Wilson polynomials. These asymptotic expansions are in fact convergent expansions. … For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). For asymptotic approximations to the largest zeros of the q -Laguerre and continuous q - 1 -Hermite polynomials see Chen and Ismail (1998).
8: 12.16 Mathematical Applications
In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. …
9: 2.1 Definitions and Elementary Properties
§2.1(i) Asymptotic and Order Symbols
§2.1(iii) Asymptotic Expansions
Symbolically, …
§2.1(iv) Uniform Asymptotic Expansions
§2.1(v) Generalized Asymptotic Expansions
10: 24.11 Asymptotic Approximations
§24.11 Asymptotic Approximations
24.11.1 ( - 1 ) n + 1 B 2 n 2 ( 2 n ) ! ( 2 π ) 2 n ,
24.11.2 ( - 1 ) n + 1 B 2 n 4 π n ( n π e ) 2 n ,
24.11.3 ( - 1 ) n E 2 n 2 2 n + 2 ( 2 n ) ! π 2 n + 1 ,
24.11.4 ( - 1 ) n E 2 n 8 n π ( 4 n π e ) 2 n .