nonuniformity of convergence

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1: 6.16 Mathematical Applications

… ►Compare Figure 6.16.1. ►This nonuniformity of convergence is an illustration of the Gibbs phenomenon. …

2: 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 $x$.

… ►In consequence, the approximation is nonuniform with respect to $\alpha$ and deteriorates severely as $\alpha \to 0$. …

3: 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. …

4: 3.8 Nonlinear Equations

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

5: 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).

6: T. Mark Dunster

… ►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

7: 36.8 Convergent Series Expansions

§36.8 Convergent Series Expansions

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8: 1.9 Calculus of a Complex Variable

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§1.9(v) Infinite Sequences and Series

… ►Absolutely convergent series are also convergent. … ►

§1.9(vi) Power Series

… ► … ►

Dominated Convergence Theorem

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9: 1.12 Continued Fractions

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§1.12(ii) Convergents

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§1.12(iii) Existence of Convergents

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§1.12(v) Convergence

… ►Then the convergents $C_n$ satisfy …The continued fraction converges iff, in addition, …

10: 35.10 Methods of Computation

… ►These algorithms are extremely efficient, converge rapidly even for large values of $m$, and have complexity linear in $m$.