asymptotic%20behavior%20for%20large%0Avariable

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31: 33.24 Tables

… ►

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

►

•

Curtis (1964a) tabulates $P_{\ell}(\epsilon,r)$ , $Q_{\ell}(\epsilon,r)$ (§33.1), and related functions for $\ell=0,1,2$ and $\epsilon=-2(.2)2$ , with $x=0(.1)4$ for $\epsilon<0$ and $x=0(.1)10$ for $\epsilon\geq 0$ ; 6D.

…

32: 23 Weierstrass Elliptic and Modular
Functions

…

33: 2.2 Transcendental Equations

§2.2 Transcendental Equations

… ►An important case is the reversion of asymptotic expansions for zeros of special functions. … ►

2.2.7

f(x)\sim x+f_{0}+f_{1}x^{-1}+f_{2}x^{-2}+\cdots,

x\to\infty

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $f(x)$ : function and $f_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/2.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §2.2, §2.2 and Ch.2

… ►where

F_{0}=f_{0}

and

sF_{s}

(

s\geq 1

) is the coefficient of

x^{-1}

in the asymptotic expansion of

(f(x))^{s}

(Lagrange’s formula for the reversion of series). …For other examples see de Bruijn (1961, Chapter 2).

34: Bibliography L

… ►

P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright

\omega

function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

ⓘ

External Links:: ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §4.13, 2nd item, §4.48(iv).
Encodings:: BibTeX

… ►

D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

ⓘ

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.11.
Encodings:: BibTeX

… ►

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.

ⓘ

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

… ►

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

… ►

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

…

35: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

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

g_{k}

up to

g_{20}

. … ►

§5.11(iii) Ratios

… ►

36: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in

\mathbf{x}

space, across which

\Psi_{K}(\mathbf{x};k)

\Psi^{(\mathrm{U})}(\mathbf{x};k)

acquires an exponentially-small asymptotic contribution (in

k

), associated with a complex critical point of

\Phi_{K}

\Phi^{(\mathrm{U})}

. … ►The Stokes set takes different forms for

z=0

z<0

, and

z>0

. ►For

z=0

, the set consists of the two curves … ►For

z>0

the Stokes set has two sheets. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

37: 14.8 Behavior at Singularities

§14.8 Behavior at Singularities

… ►In the next three relations

\Re\mu>0

. … ►The behavior of

\mathsf{P}^{\mu}_{\nu}\left(x\right)

and

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

x\to-1+

follows from the above results and the connection formulas (14.9.8) and (14.9.10). … ►

14.8.16

{\boldsymbol{Q}^{\mu}_{-n-(1/2)}\left(x\right)\sim\frac{\pi^{1/2}\Gamma\left(% \mu+n+\frac{1}{2}\right)}{n!\Gamma\left(\mu-n+\frac{1}{2}\right)(2x)^{n+(1/2)}% }},

n=1,2,3,\dots

\mu-n+\frac{1}{2}\neq 0,-1,-2,\dots

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $x$ : real variable, $n$ : nonnegative integer and $\mu$ : general order
Referenced by:: §14.21(iii), §14.8(iii)
Permalink:: http://dlmf.nist.gov/14.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §14.8(iii), §14.8 and Ch.14

38: 9.9 Zeros

… ►

§9.9(iv) Asymptotic Expansions

►For large

k

►

9.9.6

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

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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

… ►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). …

39: 12.16 Mathematical Applications

… ►In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. …

40: 27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …