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1: Bibliography F

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C. Ferreira, J. L. López, and E. Pérez Sinusía (2013a) The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

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External Links:: ISSN 0021-9045, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §16.13, §16.13.
Encodings:: BibTeX

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C. Ferreira, J. L. López, and E. P. Sinusía (2013b) The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.

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External Links:: ISSN 1660-5446, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §16.13, §16.13.
Encodings:: BibTeX

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J. L. Fields (1973) Uniform asymptotic expansions of certain classes of Meijer

G

-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:: Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §16.22.
Encodings:: BibTeX

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J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer

G

-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:: ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt
Referenced by:: §16.22.
Encodings:: BibTeX

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L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (L. J. Slater), ZentralBlatt
Referenced by:: 3rd item, 2nd item.
Encodings:: BibTeX

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2: Bibliography L

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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.

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

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J. L. López, P. Pagola, and E. Pérez Sinusía (2013a) Asymptotics of the first Appell function

F_{1}

with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §16.13, §16.13.
Encodings:: BibTeX

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J. L. López, P. Pagola, and E. Pérez Sinusía (2013b) Asymptotics of the first Appell function

F_{1}

with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §16.13, §16.13.
Encodings:: BibTeX

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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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J. L. López and N. M. Temme (2010b) Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363 (1), pp. 197–208.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §24.16(i), §24.16(i).
Encodings:: BibTeX

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3: Bibliography U

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F. Ursell (1972) Integrals with a large parameter. Several nearly coincident saddle-points. Proc. Cambridge Philos. Soc. 72, pp. 49–65.

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External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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4: Bibliography B

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L. Baker (1992) C Mathematical Function Handbook. McGraw-Hill, Inc., New York.

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Notes:: Includes diskette containing a large collection of mathematical function software written in C; Double-precision, maximum accuracy 9S.
External Links:: ISBN 00-791-1158-0
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, § ‣ 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

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L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.5(iv).
Encodings:: BibTeX

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G. Blanch and I. Rhodes (1955) Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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External Links:: MathReview (L. Fox)
Referenced by:: 3rd item.
Encodings:: BibTeX

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5: Bibliography K

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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

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6: Bibliography O

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A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:: ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:: Document, MathReview (F. Oberhettinger), ZentralBlatt
Referenced by:: §10.19(iii), §10.21(vii).
Encodings:: BibTeX

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F. W. J. Olver (1962) Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:: MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §10.20(i), §10.41(ii), 3rd item, 3rd item.
Encodings:: BibTeX

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7: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that

\mathbf{H}_{\nu}\left(x\right)

and

\mathbf{L}_{\nu}\left(x\right)

can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

8: Bibliography C

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F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial

{L}_{n}^{\alpha}(x)

as the index

\alpha\rightarrow\infty

and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:: ISSN 0024-1318, Document, MathReview (R. A. Askey)
Referenced by:: §18.7(iii).
Encodings:: BibTeX

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9: 10.45 Functions of Imaginary Order

… ►In consequence of (10.45.5)–(10.45.7),

\widetilde{I}_{\nu}\left(x\right)

and

\widetilde{K}_{\nu}\left(x\right)

comprise a numerically satisfactory pair of solutions of (10.45.1) when

x

is large, and either

\widetilde{I}_{\nu}\left(x\right)

and

(1/\pi)\sinh\left(\pi\nu\right)\widetilde{K}_{\nu}\left(x\right)

, or

\widetilde{I}_{\nu}\left(x\right)

and

\widetilde{K}_{\nu}\left(x\right)

, comprise a numerically satisfactory pair when

x

is small, depending whether

\nu\neq 0

\nu=0

. … ►For properties of

\widetilde{I}_{\nu}\left(x\right)

and

\widetilde{K}_{\nu}\left(x\right)

, including uniform asymptotic expansions for large

\nu

and zeros, see Dunster (1990a). In this reference

\widetilde{I}_{\nu}\left(x\right)

is denoted by

(1/\pi)\sinh\left(\pi\nu\right)L_{i\nu}(x)

. …

10: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the

\mathit{3j}

\mathit{6j}

, and

\mathit{9j}

symbols, different asymptotic forms are obtained depending on which parameters are large. … ►

34.8.1

\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ j_{2}&j_{1}&l_{3}\end{Bmatrix}=(-1)^{j_{1}+j_{2}+j_{3}+l_{3}}\*\left(\frac{4}{% \pi(2j_{1}+1)(2j_{2}+1)(2l_{3}+1)\sin\theta}\right)^{\frac{1}{2}}\*\left(\cos% \left((l_{3}+\tfrac{1}{2})\theta-\tfrac{1}{4}\pi\right)+o\left(1\right)\right),

j_{1},j_{2},j_{3}\gg l_{3}\gg 1

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than, $\sin\NVar{z}$ : sine function, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $l,l_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.8 and Ch.34

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