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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: 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.
3: 32.12 Asymptotic Approximations for Complex Variables
§32.12 Asymptotic Approximations for Complex Variables
4: 8.16 Generalizations
For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …
5: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
18.29.2 Q n ( z ; a , b , c , d q ) z n ( a z - 1 , b z - 1 , c z - 1 , d z - 1 ; q ) ( z - 2 , b c , b d , c d ; q ) , n ; z , a , b , c , d , q fixed.
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).
6: 2 Asymptotic Approximations
Chapter 2 Asymptotic Approximations
7: 9.14 Incomplete Airy Functions
For information, including asymptotic approximations, computation, and applications, see Levey and Felsen (1969), Constantinides and Marhefka (1993), and Michaeli (1996).
8: 24.11 Asymptotic Approximations
§24.11 Asymptotic Approximations
24.11.4 ( - 1 ) n E 2 n 8 n π ( 4 n π e ) 2 n .
9: 25.9 Asymptotic Approximations
§25.9 Asymptotic Approximations
25.9.1 ζ ( σ + i t ) = 1 n x 1 n s + χ ( s ) 1 n y 1 n 1 - s + O ( x - σ ) + O ( y σ - 1 t 1 2 - σ ) ,
25.9.3 ζ ( 1 2 + i t ) = n = 1 m 1 n 1 2 + i t + χ ( 1 2 + i t ) n = 1 m 1 n 1 2 - i t + O ( t - 1 / 4 ) .
For other asymptotic approximations see Berry and Keating (1992), Paris and Cang (1997); see also Paris and Kaminski (2001, pp. 380–389).
10: 18.24 Hahn Class: Asymptotic Approximations
§18.24 Hahn Class: Asymptotic Approximations
Asymptotic approximations are also provided for the zeros of K n ( x ; p , N ) in various cases depending on the values of p and μ . … For asymptotic approximations for the zeros of M n ( n x ; β , c ) in terms of zeros of Ai ( x ) 9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … For asymptotic approximations to P n ( λ ) ( x ; ϕ ) as | x + i λ | , with n fixed, see Temme and López (2001). …Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.