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31: 28.11 Expansions in Series of Mathieu Functions

… ►The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip $S$. …

32: 28.14 Fourier Series

… ►converge absolutely and uniformly on all compact sets in the $z$-plane. …

33: 1.10 Functions of a Complex Variable

… ►The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. … ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. … ►where $a_n(z)$ is analytic for all $n\ge 1$, and the convergence of the product is uniform in any compact subset of $D$. …

34: 28.32 Mathematical Applications

… ►defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. …

35: Bibliography P

… ►

Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.24(ii).

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36: 1.9 Calculus of a Complex Variable

… ►

Term-by-Term Integration

►Suppose the series ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(z)$, where $f_n(z)$ is continuous, converges uniformly on every compact set of a domain $D$, that is, every closed and bounded set in $D$. … ►Suppose ${\sum }_{n=0}^{\mathrm{\infty }}{f}_n(t)$ converges uniformly in any compact interval in $(a,b)$, and at least one of the following two conditions is satisfied: …

37: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Let $\mu$ be a probability measure on the unit circle of which the support is an infinite set. … ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

38: 13.8 Asymptotic Approximations for Large Parameters

… ►as $b\to \mathrm{\infty }$, uniformly in compact $\lambda$-intervals of $(0,\mathrm{\infty })$ and compact real $a$-intervals. …

39: 18.18 Sums

… ►Moreover, the series (18.18.2) converges uniformly on any compact domain within $E$. … ►Then (18.18.2), with $z$ replaced by $x$, applies when ; moreover, the convergence is uniform on any compact interval within $(-1,1)$. … ►The convergence of the series (18.18.4) is uniform on any compact interval in $(0,\mathrm{\infty })$. … ►The convergence of the series (18.18.6) is uniform on any compact interval in $(-\mathrm{\infty },\mathrm{\infty })$. …

40: 20.5 Infinite Products and Related Results

… ►With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the $z$-plane. …