asymptotic and order symbols

♦ 11—20 of 37 matching pages ♦

(0.001 seconds)

11—20 of 37 matching pages

11: 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).

12: 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). … ►

13: 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 $a$

… ► …

14: 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)
Referenced by:: §2.3(vi).
Encodings:: BibTeX

…

15: 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

…

16: 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).

17: 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

…

18: 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

…

19: 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). …

20: 16.11 Asymptotic Expansions

§16.11 Asymptotic Expansions

►

§16.11(i) Formal Series

… ►

§16.11(ii) Expansions for Large Variable

… ►

§16.11(iii) Expansions for Large Parameters

… ►Asymptotic expansions for the polynomials

{{}_{p+2}F_{q}}\left(-r,r+a_{0},\mathbf{a};\mathbf{b};z\right)

r\to\infty

through integer values are given in Fields and Luke (1963b, a) and Fields (1965).