M-test for uniform convergence

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11: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

… ►These asymptotic expansions are in fact convergent expansions. … ►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …

12: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.

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ISSN 1364-5021, Document, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§18.15(i).

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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13: 1.8 Fourier Series

… ►(1.8.10) continues to apply if either $a$ or $b$ or both are infinite and/or $f(x)$ has finitely many singularities in $(a,b)$, provided that the integral converges uniformly (§1.5(iv)) at $a,b$, and the singularities for all sufficiently large $\lambda$. ►

§1.8(ii) Convergence

… ►Then the series (1.8.1) converges to the sum …The convergence is non-uniform, however, at points where $f(x-)\ne f(x+)$; see §6.16(i). … ►For other tests for convergence see Titchmarsh (1962b, pp. 405–410). …

14: 17.18 Methods of Computation

… ►Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. … ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

15: 18.40 Methods of Computation

… ►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … ►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to $w(x)$, as will be considered in the following paragraphs. … ►Gautschi (2004, p. 119–120) has explored the $\epsilon \to 0^{+}$ limit via the Wynn $\epsilon$-algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in $w(x)$, depending smoothly on $x^{\prime }$, for $N\approx 4000$, for an example involving first numerator Legendre OP’s. … ►Convergence is $\sim O\left(N^{-2}\right)$. … ►Further, exponential convergence in $N$, via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

16: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

17: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

18: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

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Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

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Referenced by:

§2.8(vi).

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Bisection hardly ever converges linearly. Numer. Math. 70 (1), pp. 111–118.

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ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(iii).

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19: 3.8 Nonlinear Equations

… ► … ► … ► … ►The rule converges locally and is cubically convergent. … ►The convergence of iterative methods …

20: 2.4 Contour Integrals

… ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►If this integral converges uniformly at each limit for all sufficiently large $t$, then by the Riemann–Lebesgue lemma (§1.8(i)) … ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …