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31: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

… ►For (13.21.6), (13.21.7), and extensions to asymptotic expansions and error bounds, see Olver (1997b, Chapter 12, Exs. 12.4.5, 12.4.6). … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$. … ►This reference also includes error bounds and extensions to asymptotic expansions and complex values of $x$. …

32: Bibliography H

… ►

Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(xiv).

Encodings:

BibTeX

…

33: 2.4 Contour Integrals

… ►For error bounds see Boyd (1993). … ►Additionally, it may be advantageous to arrange that $\mathrm{\Im }\left(zp(t)\right)$ is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. …

34: 2.8 Differential Equations with a Parameter

… ►For error bounds, extensions to pure imaginary or complex $u$, an extension to inhomogeneous differential equations, and examples, see Olver (1997b, Chapter 10). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$ and $u$, zeros, connection formulas, extensions to inhomogeneous equations, and examples, see Olver (1997b, Chapters 11, 13), Olver (1964b), Reid (1974a, b), Boyd (1987), and Baldwin (1991). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For results, including error bounds, see Olver (1977c). …

35: DLMF Project News

error generating summary

36: 19.27 Asymptotic Approximations and Expansions

… ►Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). …

37: 5.4 Special Values and Extrema

… ►For error bounds for this estimate see Walker (2007, Theorem 5).

38: 11.9 Lommel Functions

… ►For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …

39: 10.21 Zeros

… ►For error bounds see Wong and Lang (1990), Wong (1995), and Elbert and Laforgia (2000). … … ►An error bound is included for the case $\nu \ge \frac{3}{2}$. … ►For error bounds for (10.21.32) see Qu and Wong (1999); for (10.21.36) and (10.21.37) see Elbert and Laforgia (1997). …

40: 2.3 Integrals of a Real Variable

… ►

2.3.3 $${\sigma }_n=\underset{(0,\mathrm{\infty })}{sup}(t^{-1}\ln |q^{(n)}(t)/q^{(n)}(0)|)$$

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Defines:

${\sigma }_n$ (locally)

Symbols:

$\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $sup$: least upper bound (supremum), $q(t)$: infinitely differentiable function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/2.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(i), §2.3 and Ch.2

►is finite and bounded for $n=0,1,2,\mathrm{\dots }$, then the $n$th error term (that is, the difference between the integral and $n$th partial sum in (2.3.2)) is bounded in absolute value by $|q^{(n)}(0)/(x^n(x-{\sigma }_n))|$ when $x$ exceeds both $0$ and ${\sigma }_n$. … ►In both cases the $n$th error term is bounded in absolute value by $x^{-n}{𝒱}_{a,b}\left(q^{(n-1)}(t)\right)$, where the variational operator ${𝒱}_{a,b}$ is defined by … ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►For proofs of the results of this subsection, error bounds, and an example, see Olver (1974). …