DLMF: Untitled Document

… ►

E. Wagner (1988) Asymptotische Entwicklungen der hypergeometrischen Funktion

{F}(a,b,c,z)

für

|c|\to\infty

und konstante Werte

a, b

und

z

. Demonstratio Math. 21 (2), pp. 441–458 (German).

ⓘ

External Links:: ISSN 0420-1213, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §15.12(ii), §15.12(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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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

…

… ►

A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.

ⓘ

External Links:: ISSN 1021-2655, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §13.9(i), §15.13.
Encodings:: BibTeX

… ►

S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.

ⓘ

Notes:: Includes diskette containing a large collection of mathematical function software written in Fortran. Implementation in double precision. Maximum accuracy 16S.
External Links:: ISBN 0-471-11963-6, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: 6th item, 3rd item, 1st item, 11st item, 7th item, 2nd item, 6th item, 3rd item, 2nd item, 5th item, 8th item, 4th item, 2nd item, §19.37(ii), §19.37(iii), §19.37(iii), §22.21, §24.19(i), §28.1, item (b), item (d), 8th item, 3rd item, 6th item, §5.22(ii), §5.22(iii), 2nd item, 2nd item, 4th item, 2nd item, 2nd item, §8.17(v), 4th item, 2nd item, 5th item, 3rd item, 3rd item, 3rd item, § ‣ 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, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

►

A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.

ⓘ

External Links:: MathReview (Vadim B. Kuznetsov), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

Ya. M. Zhileĭkin and A. B. Kukarkin (1995) A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

ⓘ

Notes:: English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905
External Links:: ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

►

C. H. Ziener, M. Rückl, T. Kampf, W. R. Bauer, and H. P. Schlemmer (2012) Mathieu functions for purely imaginary parameters. J. Comput. Appl. Math. 236 (17), pp. 4513–4524.

ⓘ

External Links:: ISSN 0377-0427, Document, FullText, MathReview (Bartolomeu D. B. Figueiredo), ZentralBlatt
Referenced by:: §28.36(iii), §28.36(iii).
Encodings:: BibTeX

…

… ►If

m

is a large positive integer, then …

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

… ►If

b

and

x

are fixed, with

b>0

and

0<x<1

, then as

a\to\infty

… ►

Large $a$ , Fixed $b$

… ►Then as

a+b\to\infty

, … ►For asymptotic expansions for large values of

a

and/or

b

of the

x

-solution of the equation …

… ►

M. Nardin, W. F. Perger, and A. Bhalla (1992a) Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

ⓘ

Notes:: Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.
External Links:: ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

►

M. Nardin, W. F. Perger, and A. Bhalla (1992b) Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

ⓘ

Notes:: Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.
External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §13.32(iii).
Encodings:: BibTeX

… ►

M. Newman (1967) Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §27.15.
Encodings:: BibTeX

… ►

E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

ⓘ

Notes:: See for additions and corrections same journal, Sect. B 75, 149–163 (1975)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii), §7.21, §7.7(iii).
Encodings:: BibTeX

… ►

Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)

ⓘ

Notes:: A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.
External Links:: Home Page
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.
Encodings:: BibTeX

…

… ►These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns. …

… ►

8.12.18

\rselection{Q\left(a,z\right)\\ P\left(a,z\right)}\sim\frac{z^{a-\frac{1}{2}}e^{-z}}{\Gamma\left(a\right)}{% \left(d(\pm\chi)\sum_{k=0}^{\infty}\frac{A_{k}(\chi)}{z^{k/2}}\mp\sum_{k=1}^{% \infty}\frac{B_{k}(\chi)}{z^{k/2}}\right)},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $P\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $Q\left(\NVar{a},\NVar{z}\right)$ : normalized incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $\chi$ , $d(\pm\chi)$ , $A_{k}(\chi)$ and $B_{k}(\chi)$
Referenced by:: §8.12, Erratum (V1.0.16) for Equation (8.12.18)
Permalink:: http://dlmf.nist.gov/8.12.E18
Encodings:: pMML, png, TeX
Errata (effective with 1.0.16):: The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.
Reported 2017-01-28 by Richard Paris
See also:: Annotations for §8.12 and Ch.8

… ►

B_{1}(\chi)=\tfrac{1}{3}+\tfrac{1}{6}\chi^{2}.

►Higher coefficients

A_{k}(\chi)

,

B_{k}(\chi)

, up to

k=8

, are given in Paris (2002b). ►Lastly, a uniform approximation for

\Gamma\left(a,ax\right)

for large

a

, with error bounds, can be found in Dunster (1996a). …

… ►converges for all sufficiently large

x

, and

q(t)

is infinitely differentiable in a neighborhood of the origin. … ►When

p(t)

is real and

x

is a large positive parameter, the main contribution to the integral … ►When the parameter

x

is large the contributions from the real and imaginary parts of the integrand in … ►If

p(b)

is finite, then both endpoints contribute: … ►

k

(

\leq\infty

) and

\lambda

are positive constants,

\alpha

is a variable parameter in an interval

\alpha_{1}\leq\alpha\leq\alpha_{2}

with

\alpha_{1}\leq 0

and

0<\alpha_{2}\leq k

, and

x

is a large positive parameter. …

… ►

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

…

… ►

B. Deconinck and M. van Hoeij (2001) Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

ⓘ

External Links:: ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

… ►

B. I. Dunlap and B. R. Judd (1975) Novel identities for simple

n

-

j

symbols. J. Mathematical Phys. 16, pp. 318–319.

ⓘ

External Links:: Document, MathReview (M. J. Westwater)
Referenced by:: §34.5(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1990a) Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

ⓘ

Notes:: Errata: In eq. (2.8) replace $(x^{2}/4)^{2}$ by $(x^{2}/4)^{s}$ . In eq. (4.2) $\phi(\zeta)$ should be replaced by $\psi(\zeta)$ . In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$ . In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.
External Links:: ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.24, §10.45, §2.8(iv).
Encodings:: BibTeX

… ►

T. M. Dunster (1991) Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.

ⓘ

External Links:: ISSN 0308-2105, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.20(i), §14.20(ix), §14.20(ix), §14.20(viii), §14.23.
Encodings:: BibTeX

… ►

T. M. Dunster (2003b) Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.

ⓘ

External Links:: ISSN 0308-2105, Document, MathReview, ZentralBlatt
Referenced by:: §14.15(i), §14.15(i), §14.15(ii), §14.15(ii), §14.26.
Encodings:: BibTeX

…

large a or b

11—20 of 566 matching pages

11: Bibliography W

12: Bibliography Z

13: 10.70 Zeros

14: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

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

Large $a$ , Fixed $b$

15: Bibliography N

16: About the Project

17: 8.12 Uniform Asymptotic Expansions for Large Parameter

18: 2.3 Integrals of a Real Variable

19: Bibliography L

20: Bibliography D

large a or b

11—20 of 566 matching pages

11: Bibliography W

12: Bibliography Z

13: 10.70 Zeros

14: 8.18 Asymptotic Expansions of I x ⁡ ( a , b )

§8.18 Asymptotic Expansions of I x ⁡ ( a , b )

Large a , Fixed b

15: Bibliography N

16: About the Project

17: 8.12 Uniform Asymptotic Expansions for Large Parameter

18: 2.3 Integrals of a Real Variable

19: Bibliography L

20: Bibliography D

14: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

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

Large $a$ , Fixed $b$