asymptotic behavior

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31—35 of 35 matching pages

31: 8.13 Zeros

… ►When

-5\leq a\leq 4

the behavior of the

x

-zeros as functions of

a

can be seen by taking the slice

\gamma^{*}\left(a,x\right)=0

of the surface depicted in Figure 8.3.6. … ►For asymptotic approximations for

x_{+}(a)

and

x_{-}(a)

a\to-\infty

see Tricomi (1950b), with corrections by Kölbig (1972b). For more accurate asymptotic approximations see Thompson (2012). …

32: 2.11 Remainder Terms; Stokes Phenomenon

… ►

§2.11(i) Numerical Use of Asymptotic Expansions

… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ► … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the

F

-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the

F

-functions. … ► …

33: 14.20 Conical (or Mehler) Functions

… ►

§14.20(iii) Behavior as $x\to 1$

►The behavior of

\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(\pm x\right)

x\to 1-

is given in §14.8(i). … ►

§14.20(vii) Asymptotic Approximations: Large $\tau$ , Fixed $\mu$

… ►For asymptotic expansions and explicit error bounds, see Olver (1997b, pp. 473–474). … ►

§14.20(viii) Asymptotic Approximations: Large $\tau$ , $0\leq\mu\leq A\tau$

…

34: 18.2 General Orthogonal Polynomials

… ►For OP’s

p_{n}(x)

with weight function in the class

\mathcal{G}

there are asymptotic formulas as

n\to\infty

, respectively for

x\in\mathbb{C}

outside

[-1,1]

and for

x\in[-1,1]

, see Szegő (1975, Theorems 12.1.2, 12.1.4). …This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with

p_{n}(x)=T_{n}\left(x\right)

, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►If these

x_{k}

satisfy

\sum_{k}(|x_{k}|-1)^{\ifrac{1}{2}}<\infty

then Szegő type asymptotics outside

[-1,1]

can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following). …

35: Errata

… ►

Chapter 19

Factors inside square roots on the right-hand sides of formulas (19.18.6), (19.20.10), (19.20.19), (19.21.7), (19.21.8), (19.21.10), (19.25.7), (19.25.10) and (19.25.11) were written as products to ensure the correct multivalued behavior.

Reported by Luc Maisonobe on 2021-06-07

… ►

Notation

The symbol $\sim$ is used for two purposes in the DLMF, in some cases for asymptotic equality and in other cases for asymptotic expansion, but links to the appropriate definitions were not provided. In this release changes have been made to provide these links.

►

Subsection 2.1(iii)

A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

►

Equation (2.11.4)

Because (2.11.4) is not an asymptotic expansion, the symbol $\sim$ that was used originally is incorrect and has been replaced with $\approx$ , together with a slight change of wording.

►

Equation (13.9.16)

Originally was expressed in term of asymptotic symbol $\sim$ . As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$ .

…