G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

â“˜

External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

… â–º

G. Nemes (2014b) The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

â“˜

External Links:: ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

â–º

G. Nemes (2014c) The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

â“˜

External Links:: ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: (11.11.10), (11.11.8), §11.11(iii).
Encodings:: BibTeX

… â–º

G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

â“˜

External Links:: ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt
Referenced by:: (9.7.16), (9.7.17), §9.7(iii), §9.7(iv).
Encodings:: BibTeX

… â–º

J. J. Nestor (1984) Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

â“˜

Referenced by:: §2.8(vi).
Encodings:: BibTeX

…

23: Bibliography T

… â–º

N. M. Temme (1978) Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

â“˜

External Links:: ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §13.8(ii), §13.8(ii).
Encodings:: BibTeX

… â–º

N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

â“˜

External Links:: ISSN 0033-569X, FullText, MathReview (I. FenyÅ‘), ZentralBlatt
Referenced by:: §2.4(i).
Encodings:: BibTeX

… â–º

N. M. Temme (1995c) Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

â“˜

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

… â–º

N. M. Temme (1997) Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

â“˜

External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §10.19(iii), §10.20(i), §10.74(i).
Encodings:: BibTeX

… â–º

N. M. Temme (2022) Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

â“˜

External Links:: ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §13.8(iv).
Encodings:: BibTeX

…

24: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

â–º

§13.20(i) Large $\mu$ , Fixed $\kappa$

… â–º … â–º

§13.20(v) Large $\mu$ , Other Expansions

… â–º

25: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

… â–ºFor large

\beta

, fixed

\alpha

, and

0\leq n/\beta\leq c

, Dunster (1999) gives asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

that are uniform in unbounded complex

z

-domains containing

z=\pm 1

. …This reference also supplies asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

for large

n

, fixed

\alpha

, and

0\leq\beta/n\leq c

. The latter expansions are in terms of Bessel functions, and are uniform in complex

z

-domains not containing neighborhoods of 1. … â–ºThe first term of this expansion also appears in SzegÅ‘ (1975, Theorem 8.21.7). …

26: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… â–º

§28.8(ii) Sips’ Expansions

… â–º

Barrett’s Expansions

… â–º

Dunster’s Approximations

… â–º

27: Bibliography J

… â–º

X.-S. Jin and R. Wong (1998) Uniform asymptotic expansions for Meixner polynomials. Constr. Approx. 14 (1), pp. 113–150.

â“˜

External Links:: ISSN 0176-4276, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §18.24, §2.4(vi).
Encodings:: BibTeX

… â–º

D. S. Jones (1972) Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

â“˜

External Links:: Document, MathReview (A. Erdelyi), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

… â–º

D. S. Jones (1997) Introduction to Asymptotics: A Treatment Using Nonstandard Analysis. World Scientific Publishing Co. Inc., River Edge, NJ.

â“˜

External Links:: ISBN 981-02-2915-1, MathReview (W. A. J. Luxemburg), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

… â–º

D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

â“˜

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §12.10(vi).
Encodings:: BibTeX

… â–º

S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions

\overline{ps}\,^{r}_{n}(\eta,h)

and

\overline{qs}\,^{r}_{n}(\eta,h)

for large

h

. Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

â“˜

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §30.9(ii).
Encodings:: BibTeX

…

28: Bibliography W

… â–º

Z. Wang and R. Wong (2002) Uniform asymptotic expansion of

J_{\nu}(\nu a)

via a difference equation. Numer. Math. 91 (1), pp. 147–193.

â“˜

External Links:: ISSN 0029-599X, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.20(ii).
Encodings:: BibTeX

… â–º

R. Wong and Y.-Q. Zhao (2003) Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

â“˜

External Links:: ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.15(i), §18.15(i).
Encodings:: BibTeX

… â–º

R. Wong and Y. Zhao (2004) Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.

â“˜

External Links:: ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt
Referenced by:: §18.15(i).
Encodings:: BibTeX

… â–º

R. Wong (1973a) An asymptotic expansion of

{W}_{k,\,m}(z)

with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

â“˜

External Links:: FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt
Referenced by:: §13.19.
Encodings:: BibTeX

â–º

R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

â“˜

External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §8.11(v).
Encodings:: BibTeX

…

29: 24.16 Generalizations

… â–º

§24.16(i) Higher-Order Analogs

â–º

Polynomials and Numbers of Integer Order

â–ºFor

\ell=0,1,2,\dotsc

, Bernoulli and Euler polynomials of order $\ell$ are defined respectively by … â–ºFor extensions of

B^{(\ell)}_{n}\left(x\right)

to complex values of

x

n

, and

\ell

, and also for uniform asymptotic expansions for large

x

and large

n

, see Temme (1995b) and López and Temme (1999b, 2010b). … â–ºIn no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989));

p

-adic integer order Bernoulli numbers (Adelberg (1996));

p

-adic

q

-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

30: 2.11 Remainder Terms; Stokes Phenomenon

… â–º

§2.11(i) Numerical Use of Asymptotic Expansions

… â–º â–ºIn order to guard against this kind of error remaining undetected, the wanted function may need to be computed by another method (preferably nonasymptotic) for the smallest value of the (large) asymptotic variable

x

that is intended to be used. … â–ºFor second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). … â–º …

uniform asymptotic expansions for large order

21—30 of 38 matching pages

21: 8.18 Asymptotic Expansions of I x â¡ ( a , b )

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Symmetric Case

General Case

Inverse Function

22: Bibliography N

23: Bibliography T

24: 13.20 Uniform Asymptotic Approximations for Large Î¼

§13.20 Uniform Asymptotic Approximations for Large Î¼

§13.20(i) Large Î¼ , Fixed Îº

§13.20(v) Large Î¼ , Other Expansions

25: 18.15 Asymptotic Approximations

§18.15 Asymptotic Approximations

26: 28.8 Asymptotic Expansions for Large q

§28.8 Asymptotic Expansions for Large q

§28.8(ii) Sips’ Expansions

Barrett’s Expansions

Dunster’s Approximations

27: Bibliography J

28: Bibliography W

29: 24.16 Generalizations

§24.16(i) Higher-Order Analogs

Polynomials and Numbers of Integer Order

30: 2.11 Remainder Terms; Stokes Phenomenon

§2.11(i) Numerical Use of Asymptotic Expansions

21: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

24: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20(i) Large $\mu$ , Fixed $\kappa$

§13.20(v) Large $\mu$ , Other Expansions

26: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$