asymptotic and order symbols

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11: 2.11 Remainder Terms; Stokes Phenomenon

… ► ►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. … ►For second-order differential equations, see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Murphy and Wood (1997). ►For higher-order differential equations, see Olde Daalhuis (1998a, b). … ► …

12: 15.12 Asymptotic Approximations

§15.12 Asymptotic Approximations

►

§15.12(i) Large Variable

… ►

§15.12(ii) Large $c$

… ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

13: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where

\delta

denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►With

x=1

, an asymptotic expansion of

e_{n}(nx)/e^{nx}

follows from (8.11.14) and (8.11.16). … ►

14: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

… ►

§13.8(ii) Large $b$ and $z$ , Fixed $a$ and $b/z$

… ►For other asymptotic expansions for large

b

and

z

see López and Pagola (2010). … ►

§13.8(iii) Large $a$

… ► …

15: Bibliography F

… ►

J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

ⓘ

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

►

J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

ⓘ

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

… ►

J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

ⓘ

External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

… ►

J. P. M. Flude (1998) The Edmonds asymptotic formulas for the

3j

and

6j

symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt
Referenced by:: §34.8.
Encodings:: BibTeX

… ►

C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

ⓘ

External Links:: ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt
Referenced by:: §2.3(vi).
Encodings:: BibTeX

…

16: Bibliography D

… ►

T. M. Dunster, R. B. Paris, and S. Cang (1998) On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5 (3), pp. 223–247.

ⓘ

External Links:: ISSN 1073-2772, Pdf, MathReview (John G. Stalker), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

… ►

T. M. Dunster (1990b) Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §14.26, §2.8(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1994b) Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt
Referenced by:: §2.8(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

ⓘ

External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.8(vi), §8.12, §8.12.
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

…

17: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

… ►

13.7.1

{\mathbf{M}}\left(a,b,x\right)\sim\frac{e^{x}x^{a-b}}{\Gamma\left(a\right)}% \sum_{s=0}^{\infty}\frac{{\left(1-a\right)_{s}}{\left(b-a\right)_{s}}}{s!}x^{-% s},

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/13.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.7(i), §13.7 and Ch.13

… ►

§13.7(ii) Error Bounds

… ►

§13.7(iii) Exponentially-Improved Expansion

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

18: Bibliography B

… ►

C. B. Balogh (1967) Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

ⓘ

External Links:: ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.45.
Encodings:: BibTeX

… ►

W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

ⓘ

External Links:: ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

… ►

M. V. Berry and C. J. Howls (1993) Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality. Proc. Roy. Soc. London Ser. A 443, pp. 107–126.

ⓘ

External Links:: ISSN 0962-8444, FullText, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §36.12(i), §36.12(ii), §36.12(iii).
Encodings:: BibTeX

… ►

J. P. Boyd (1998) Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Mathematics and its Applications, Vol. 442, Kluwer Academic Publishers, Boston-Dordrecht.

ⓘ

External Links:: ISBN 0-7923-5072-3, MathReview (Alan Jeffrey), ZentralBlatt
Referenced by:: §22.19(iii).
Encodings:: BibTeX

… ►

W. G. C. Boyd (1990a) Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

ⓘ

External Links:: MathReview (John S. Lew), ZentralBlatt
Referenced by:: §2.8(iv).
Encodings:: BibTeX

…

19: Bibliography L

… ►

S. Lai and Y. Chiu (1990) Exact computation of the

3

j

and

6

j

symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

ⓘ

Notes:: A Fortran program is included.
External Links:: Document, ZentralBlatt
Referenced by:: §34.15.
Encodings:: BibTeX

►

S. Lai and Y. Chiu (1992) Exact computation of the

9

j

symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

ⓘ

Notes:: Double-precision Fortran
External Links:: ISSN 0010-4655, Document, Code, MathReview
Referenced by:: 7th item.
Encodings:: BibTeX

… ►

H. A. Lauwerier (1974) Asymptotic Analysis. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.

ⓘ

External Links:: MathReview (C. Comstock), ZentralBlatt
Referenced by:: Ch.2.
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

… ►

E. R. Love (1972a) Addendum to: “Changing the order of integration”. J. Austral. Math. Soc. 14, pp. 383–384.

ⓘ

External Links:: Document, MathReview (G. W. Johnson), ZentralBlatt
Referenced by:: §1.5(v).
Encodings:: BibTeX

…

20: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

►With

\psi\left(x\right)

denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of

K\left(k\right)

and

E\left(k\right)

near the singularity at

k=1

is given by the following convergent series: ►

19.12.1

K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),

0<|k^{\prime}|<1

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

… ►For the asymptotic behavior of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

\phi\to\tfrac{1}{2}\pi-

and

k\to 1-

see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for

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

, with different variables, are given in Karp et al. (2007). …