DLMF: Untitled Document

§11.6 Asymptotic Expansions

… ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … ►Here …

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Olver (1960) tabulates $j_{n,m}$ , $J_{n}'\left(j_{n,m}\right)$ , ${j^{\prime}_{n,m}}$ , $J_{n}\left({j^{\prime}_{n,m}}\right)$ , $y_{n,m}$ , $Y_{n}'\left(y_{n,m}\right)$ , ${y^{\prime}_{n,m}}$ , $Y_{n}\left({y^{\prime}_{n,m}}\right)$ , $n=0(\tfrac{1}{2})20\tfrac{1}{2}$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ ; see §10.21(viii), and more fully Olver (1954).

… ►

•

Bickley et al. (1952) tabulates $x^{-n}I_{n}\left(x\right)$ or $e^{-x}I_{n}\left(x\right)$ , $x^{n}K_{n}\left(x\right)$ or $e^{x}K_{n}\left(x\right)$ , $n=2(1)20$ , $x=0$ (.01 or .1) 10(.1) 20, 8S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20$ , $x=0$ or $0.1(.1)20$ , 10S.

… ►

•

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_{n}\left(z\right)$ and $K_{n}'\left(z\right)$ , for $n=2(1)20$ , 9S.

… ►

•

Olver (1960) tabulates $a^{\prime}_{n,m}$ , $\mathsf{j}_{n}\left(a^{\prime}_{n,m}\right)$ , $b^{\prime}_{n,m}$ , $\mathsf{y}_{n}\left(b^{\prime}_{n,m}\right)$ , $n=1(1)20$ , $m=1(1)50$ , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as $n\to\infty$ .

…

… ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►

§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). … ►

12.11.4

u_{a,s}\sim 2^{\frac{1}{2}}\mu\left(p_{0}(\alpha)+\frac{p_{1}(\alpha)}{\mu^{4}% }+\frac{p_{2}(\alpha)}{\mu^{8}}+\cdots\right),

ⓘ

Defines:: $u_{a,s}$ : zeros (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $s$ : nonnegative integer, $a$ : real or complex parameter, $\alpha$ and $p_{n}(\zeta)$ : coefficients
Referenced by:: §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

… ►

12.11.9

u_{a,1}\sim 2^{\frac{1}{2}}\mu\left(1-1.85575\;708\mu^{-4/3}-0.34438\;34\mu^{-% 8/3}-0.16871\;5\mu^{-4}-0.11414\mu^{-16/3}-0.0808\mu^{-20/3}-\cdots\right),

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $a$ : real or complex parameter and $u_{a,s}$ : zeros
Referenced by:: §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

…

… ►

D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

►

D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

ⓘ

External Links:: ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

►

D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

ⓘ

External Links:: ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

… ►

W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
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

…

… ► … ►

§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ► … ►For example, using double precision

d_{20}

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

§16.22 Asymptotic Expansions

►Asymptotic expansions of

{G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)

for large

z

are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer

G

-functions with large parameters see Fields (1973, 1983).

§5.11 Asymptotic Expansions

►

§5.11(i) Poincaré-Type Expansions

… ►and … ►Wrench (1968) gives exact values of

g_{k}

up to

g_{20}

. … ►

§28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►

§28.8(ii) Sips’ Expansions

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). ►

§28.8(iii) Goldstein’s Expansions

…

… ►

§9.9(iv) Asymptotic Expansions

… ►

9.9.6

a_{k}=-T\left(\tfrac{3}{8}\pi(4k-1)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $T$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.7

\operatorname{Ai}'\left(a_{k}\right)=(-1)^{k-1}V\left(\tfrac{3}{8}\pi(4k-1)% \right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}$ , $k$ : nonnegative integer and $V$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

►

9.9.8

a^{\prime}_{k}=-U\left(\tfrac{3}{8}\pi(4k-3)\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $a^{\prime}_{\NVar{k}}$ : $k$ th zero of Airy $\operatorname{Ai}'$ , $k$ : nonnegative integer and $U$ : expansion
Source:: Olver (1954, (A 20))
Permalink:: http://dlmf.nist.gov/9.9.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.9(iv), §9.9 and Ch.9

… ►For error bounds for the asymptotic expansions of

a_{k}

,

b_{k}

,

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

… ►

FDLIBM (free C library)

ⓘ

Notes:: The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.
External Links:: GAMS (package FDLIBM)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

ⓘ

External Links:: MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §6.15.
Encodings:: BibTeX

… ►

H. E. Fettis and J. C. Caslin (1964) Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

ⓘ

Notes:: Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.
Referenced by:: §19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).
Encodings:: BibTeX

… ►

J. L. Fields and J. Wimp (1961) Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

ⓘ

External Links:: FullText, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.10.
Encodings:: BibTeX

… ►

G. Freud (1969) On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

ⓘ

External Links:: ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)
Referenced by:: §18.32.
Encodings:: BibTeX

…

Mittag-Leffler%20expansion

11—20 of 341 matching pages

11: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

12: 10.75 Tables

13: 12.11 Zeros

§12.11(ii) Asymptotic Expansions of Large Zeros

§12.11(iii) Asymptotic Expansions for Large Parameter

14: Bibliography N

15: 2.11 Remainder Terms; Stokes Phenomenon

§2.11(iii) Exponentially-Improved Expansions

16: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

17: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

§5.11(i) Poincaré-Type Expansions

18: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

§28.8(ii) Sips’ Expansions

§28.8(iii) Goldstein’s Expansions

19: 9.9 Zeros

§9.9(iv) Asymptotic Expansions

20: Bibliography F

Mittag-Leffler%20expansion

11—20 of 341 matching pages

11: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

12: 10.75 Tables

13: 12.11 Zeros

§12.11(ii) Asymptotic Expansions of Large Zeros

§12.11(iii) Asymptotic Expansions for Large Parameter

14: Bibliography N

15: 2.11 Remainder Terms; Stokes Phenomenon

§2.11(iii) Exponentially-Improved Expansions

16: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

17: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

§5.11(i) Poincaré-Type Expansions

18: 28.8 Asymptotic Expansions for Large q

§28.8 Asymptotic Expansions for Large q

§28.8(ii) Sips’ Expansions

§28.8(iii) Goldstein’s Expansions

19: 9.9 Zeros

§9.9(iv) Asymptotic Expansions

20: Bibliography F

18: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$