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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^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)

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^{-\mu}_{\nu}\left(x\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

for

1<x<\infty

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)\sim x,

x\to\infty

ⓘ

Symbols:: $\sim$ : asymptotic equality and $f(x)$ : function
Referenced by:: §2.2
Permalink:: http://dlmf.nist.gov/2.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2 and Ch.2

… ►An important case is the reversion of asymptotic expansions for zeros of special functions. …where

F_{0}=f_{0}

and

sF_{s}

(

s\geq 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\to\infty

, 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_{2n}\sim\frac{2(2n)!}{(2\pi)^{2n}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.2

(-1)^{n+1}B_{2n}\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.3

(-1)^{n}E_{2n}\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

►

24.11.4

(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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