# compact set

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## 1—10 of 22 matching pages

##### 1: 28.14 Fourier Series

##### 2: 21.2 Definitions

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►This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\mathbf{\Omega}$ spaces; hence $\theta \left(\mathbf{z}\right|\mathbf{\Omega})$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\mathbf{\Omega}$.
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##### 3: 20.5 Infinite Products and Related Results

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►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.
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##### 4: 1.9 Calculus of a Complex Variable

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###### 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 $D$. …##### 5: 23.2 Definitions and Periodic Properties

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►The double series and double product are absolutely and uniformly convergent in compact sets in $\u2102$ that do not include lattice points.
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##### 6: 28.4 Fourier Series

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►The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane.
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##### 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

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►The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$-plane.
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##### 8: 1.16 Distributions

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►If the support of $\varphi $ is a compact set (§1.9(vii)), then $\varphi $ is called a

*function of compact support*. … ►A sequence $\{{\varphi}_{n}\}$ of test functions*converges*to a test function $\varphi $ if the support of every ${\varphi}_{n}$ is contained in a fixed compact set $K$ and as $n\to \mathrm{\infty}$ the sequence $\{{\varphi}_{n}^{(k)}\}$ converges uniformly on $K$ to ${\varphi}^{(k)}$ for $k=0,1,2,\mathrm{\dots}$. … ►Let $\mathcal{D}({\mathbb{R}}^{n})={\mathcal{D}}_{n}$ be the set of all infinitely differentiable functions in $n$ variables, $\varphi ({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$, with compact support in ${\mathbb{R}}^{n}$. …##### 9: 1.10 Functions of a Complex Variable

##### 10: 21.7 Riemann Surfaces

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►In almost all applications, a Riemann theta function is associated with a compact Riemann surface.
…Equation (21.7.1) determines a plane algebraic curve in ${\u2102}^{2}$, which is made compact by adding its points at infinity.
…This compact curve may have singular points, that is, points at which the gradient of $\stackrel{~}{P}$ vanishes.
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*In this way, we associate a Riemann theta function with every compact Riemann surface $\mathrm{\Gamma}$.*… ►Then the*prime form*on the corresponding compact Riemann surface $\mathrm{\Gamma}$ is defined by …