M-test for uniform convergence

SearchAdvancedHelp

(0.002 seconds)

21—30 of 207 matching pages

21: 3.11 Approximation Techniques

… ►

§3.11(i) Minimax Polynomial Approximations

… ►The iterative process converges locally and quadratically (§3.8(i)). … ►converges uniformly. … ►Then the minimax (or best uniform) rational approximation … ► …

22: 16.2 Definition and Analytic Properties

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where …

23: 13.27 Mathematical Applications

… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).

24: 14.15 Uniform Asymptotic Approximations

§14.15 Uniform Asymptotic Approximations

►

§14.15(i) Large $\mu$, Fixed $\nu$

… ►In other words, the convergent hypergeometric series expansions of ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu \to \mathrm{\infty }$, with scale $1/\mathrm{\Gamma }\left(j+1+\mu \right)$, $j=0,1,2,\mathrm{\dots }$; compare §2.1(v). … ►For convergent series expansions see Dunster (2004). … ►

25: Bibliography B

… ►

Application of uniform asymptotics to the second Painlevé transcendent. Arch. Rational Mech. Anal. 143 (3), pp. 241–271.

ⓘ

External Links:

ISSN 0945-8396, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

… ►

Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§36.14(iii), §9.16.

Encodings:

BibTeX

►

Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

ⓘ

Referenced by:

§36.14(i), §9.16.

Encodings:

BibTeX

… ►

Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

ⓘ

External Links:

ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

ⓘ

External Links:

ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

…

26: 3.9 Acceleration of Convergence

§3.9 Acceleration of Convergence

… ► … ►provided that the right-hand side converges. … ►Examples are provided by the following analytic transformations of slowly-convergent series into rapidly convergent ones: … ►For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).

27: 2.3 Integrals of a Real Variable

… ►converges for all sufficiently large $x$, and $q(t)$ is infinitely differentiable in a neighborhood of the origin. … ►Assume again that the integral (2.3.1) converges for all sufficiently large $x$, but now … ►provided that the integral on the left-hand side of (2.3.9) converges for all sufficiently large values of $x$. … ►

The integral (2.3.13) converges absolutely for all sufficiently large $x$.

… ►A uniform approximation can be constructed by quadratic change of integration variable: …

28: 2.8 Differential Equations with a Parameter

… ►The expansions (2.8.11) and (2.8.12) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.15) and (2.8.16) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.25) and (2.8.26) are both uniform and differentiable with respect to $\xi$. … ►The expansions (2.8.29) and (2.8.30) are both uniform and differentiable with respect to $\xi$. … ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. …

29: 20.12 Mathematical Applications

… ►

§20.12(ii) Uniformization and Embedding of Complex Tori

… ►Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

30: Bibliography S

… ►

Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

ⓘ

External Links:

MathReview (J. Kuntzmann), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

… ►

Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:

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

Referenced by:

§28.8(iv).

Encodings:

BibTeX

►

Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

ⓘ

External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

ⓘ

External Links:

MathReview (A. K. Agarwal), ZentralBlatt

Referenced by:

§13.21(iv).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

ⓘ

External Links:

Abstract

Referenced by:

§18.16(v).

Encodings:

BibTeX

…