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21: 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.

ⓘ

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.

ⓘ

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.

ⓘ

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

… ►

J. L. López and P. J. Pagola (2010) The confluent hypergeometric functions

M(a,b;z)

and

U(a,b;z)

for large

b

and

z

. J. Comput. Appl. Math. 233 (6), pp. 1570–1576.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §13.8(ii), §13.8(ii).
Encodings:: BibTeX

… ►

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

ⓘ

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

…

22: Bibliography I

… ►

IMSL (commercial C, Fortran, and Java libraries) IMSL Nuerical Libraries..

ⓘ

Notes:: The IMSL Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.
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, § ‣ 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.
Encodings:: BibTeX

… ►

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… ►

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

ⓘ

External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

►

A. R. Its and A. A. Kapaev (1987) The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

ⓘ

Notes:: In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207
External Links:: ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

… ►

A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt
Referenced by:: §32.11(v).
Encodings:: BibTeX

…

23: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►where, in the notation of (19.19.7) with

a=-\frac{1}{2}

and

n=3

, … ►A summary for

F\left(\phi,k\right)

is given in Gautschi (1975, §3). … ►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)). … ►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). …

24: 11.6 Asymptotic Expansions

… ►

§11.6(i) Large $|z|$ , Fixed $\nu$

… ►

§11.6(ii) Large $|\nu|$ , Fixed $z$

►

11.6.5

\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},

|\operatorname{ph}\nu|\leq\pi-\delta

ⓘ

Symbols:: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant
Referenced by:: (11.11.7)
Permalink:: http://dlmf.nist.gov/11.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(ii), §11.6 and Ch.11

… ►

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

… ►

c_{3}(\lambda)=20\lambda^{-6}-4\lambda^{-4},

…

25: 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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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

… ►

T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

ⓘ

External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

…

26: 12.11 Zeros

… ►If

a>-\tfrac{1}{2}

, then

V\left(a,x\right)

has no positive real zeros, and if

a=\tfrac{3}{2}-2n

n\in\mathbb{Z}

, then

V\left(a,x\right)

has a zero at

x=0

. ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When

a>-\frac{1}{2}

the zeros are asymptotically given by

z_{a,s}

and

\overline{z_{a,s}}

, where

s

is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of

a

the real zeros of

U\left(a,x\right)

U'\left(a,x\right)

V\left(a,x\right)

, and

V'\left(a,x\right)

can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

27: 9.18 Tables

… ►

•

Miller (1946) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ for $x=-20(.01)2;$ $\operatorname{log}_{10}\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)/\operatorname{Ai}\left(x\right)$ for $x=0(.1)25(1)75$ ; $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=-10(.1)2.5$ ; $\operatorname{log}_{10}\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)/\operatorname{Bi}\left(x\right)$ for $x=0(.1)10$ ; $M\left(x\right)$ , $N\left(x\right)$ , $\theta\left(x\right)$ , $\phi\left(x\right)$ (respectively $F(x)$ , $G(x)$ , $\chi(x)$ , $\psi(x)$ ) for $x=-80(1)-30(.1)0$ . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

… ►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

►

•

Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

…

28: 2.11 Remainder Terms; Stokes Phenomenon

… ►with

m

a large integer. … ►In 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. … ►However, on combining (2.11.6) with the connection formula (8.19.18), with

m=1

, we derive … ►For large

\rho

the integrand has a saddle point at

t=e^{-i\theta}

. … ►For example, using double precision

d_{20}

is found to agree with (2.11.31) to 13D. …

29: 30.9 Asymptotic Approximations and Expansions

… ►

§30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of

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

and

a^{m}_{n,k}(\gamma^{2})

n\to\infty

in descending powers of

2n+1

is derived in Meixner (1944). 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).

30: 3.4 Differentiation

… ►where

A_{k}^{n}

is as in (3.3.10). … ►

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!

. …