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11: Bibliography W

… ►

R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.

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Notes:: Translated from the 1989 Japanese original. With 1 IBM-PC floppy disk (5.25 inch) containing a large general purpose mathematical software collection written in Fortran, including some special function codes. Double precision.
External Links:: ISBN 0-444-82000-0, MathReview, ZentralBlatt
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

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External Links:: ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §29.7(ii).
Encodings:: BibTeX

… ►

J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:: Document, ZentralBlatt
Referenced by:: §33.5(ii).
Encodings:: BibTeX

… ►

R. Wong (1973a) An asymptotic expansion of

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

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

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

…

12: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►

§12.10(vi) Modifications of Expansions in Elementary Functions

… ► … ►

Modified Expansions

… ►

13: Bibliography L

… ►

R. E. Langer (1934) The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:: ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

H. T. Lau (1995) A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.

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Notes:: With 1 IBM-PC floppy disk (3.5 inch; HD) containing a large general purpose mathematical software collection written in C including a collection of special function classes. The functions are computed for real variables only. (See Lau (2004) for Java version).
External Links:: ISBN 0-8493-7376-X, MathReview, ZentralBlatt
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, H. T. Lau (2004).
Encodings:: BibTeX

►

H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.

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Notes:: With 1 CD-ROM (Windows, Macintosh and Unix) containing a large general purpose mathematical software collection written in Java including a collection of special function classes. The functions are computed for real variables only. (See Lau (1995) for C version).
External Links:: ISBN 1-58488-430-4, MathReview Entry, ZentralBlatt
Referenced by:: H. T. Lau (1995).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

J. L. López (1999) Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:: Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II
External Links:: ISSN 1073-2772, Pdf, MathReview, ZentralBlatt
Referenced by:: §13.21(i), §13.24(ii).
Encodings:: BibTeX

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14: 11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$ , Fixed $\nu$

… ►

§11.6(ii) Large $|\nu|$ , Fixed $z$

… ►

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

… ►Here …These and higher coefficients

c_{k}(\lambda)

can be computed via the representations in Nemes (2015b). …

15: 19.36 Methods of Computation

… ►Here

R_{C}

is computed either by the duplication algorithm in Carlson (1995) or via (19.2.19). … ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►The function

\operatorname{cel}\left(k_{c},p,a,b\right)

is computed by successive Bartky transformations (Bulirsch and Stoer (1968), Bulirsch (1969b)). … ►The cases

k_{c}^{2}/2\leq p<\infty

and

-\infty <mrow> <mrow> <mo>−</mo> <mi mathvariant=

require different treatment for numerical purposes, and again precautions are needed to avoid cancellations. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

16: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►

§28.8(ii) Sips’ Expansions

… ►

§28.8(iii) Goldstein’s Expansions

… ►

Barrett’s Expansions

… ►

17: Bibliography C

Bibliography C

… ►

R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:: ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt
Referenced by:: §26.8(vii).
Encodings:: BibTeX

►

J. Chen (1966) On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

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External Links:: MathReview (T. Hayashida)
Referenced by:: §27.13(ii).
Encodings:: BibTeX

… ►

M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §13.29(v), §15.19(v).
Encodings:: BibTeX

… ►

M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

…

18: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

U. J. Knottnerus (1960) Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

… ►

C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively

q

-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:: Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.
External Links:: Code, Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

…

19: 30.9 Asymptotic Approximations and Expansions

… ►

§30.9(i) Prolate Spheroidal Wave Functions

… ►where … ►The cases of large

m

, and of large

m

and large

|\gamma|

, are studied in Abramowitz (1949). …The behavior of

\lambda^{m}_{n}\left(\gamma^{2}\right)

for complex

\gamma^{2}

and large

|\lambda^{m}_{n}\left(\gamma^{2}\right)|

is investigated in Hunter and Guerrieri (1982).

20: 3.4 Differentiation

… ►where

c_{n}

is defined by (3.3.12), with numerical values as in §3.3(ii). … ►

B_{-2}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),

… ►where

C

is a simple closed contour described in the positive rotational sense such that

C

and its interior lie in the domain of analyticity of

f

, and

x_{0}

is interior to

C

. Taking

C

to be a circle of radius

r

centered at

x_{0}

, we obtain … ►With the choice

r=k

(which is crucial when

k

is large because of numerical cancellation) the integrand equals

e^{k}

at the dominant points

\theta=0,2\pi

, and in combination with the factor

k^{-k}

in front of the integral sign this gives a rough approximation to

1/k!

. …