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21: 9.17 Methods of Computation

… ►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

22: 11.13 Methods of Computation

… ►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

23: 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). …

24: 6.16 Mathematical Applications

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

25: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►The series (28.19.2) converges absolutely and uniformly on compact subsets within $S$. …

26: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous $2\pi$-periodic function $f(x)$ whose second derivative is square-integrable over the interval $[0,2\pi ]$ can be expanded in a uniformly and absolutely convergent series …

27: 29.20 Methods of Computation

… ►The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to \mathrm{\infty }$. …

28: 13.31 Approximations

… ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). … ►In Luke (1977a) the following rational approximation is given, together with its rate of convergence. …

29: 28.14 Fourier Series

… ►converge absolutely and uniformly on all compact sets in the $z$-plane. … ►The rate of convergence is indicated by …

30: 35.2 Laplace Transform

… ►Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ is $0$. …