# compact set

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## 1—10 of 22 matching pages

##### 1: 28.14 Fourier Series
converge absolutely and uniformly on all compact sets in the $z$-plane. …
##### 2: 21.2 Definitions
This $g$-tuple Fourier series converges absolutely and uniformly on compact sets of the $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ spaces; hence $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is an analytic function of (each element of) $\mathbf{z}$ and (each element of) $\boldsymbol{{\Omega}}$. …
##### 3: 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. …
##### 4: 1.9 Calculus of a Complex Variable
###### Term-by-Term Integration
Suppose the series $\sum^{\infty}_{n=0}f_{n}(z)$, 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
The double series and double product are absolutely and uniformly convergent in compact sets in $\mathbb{C}$ that do not include lattice points. …
##### 6: 28.4 Fourier Series
The Fourier series of the periodic Mathieu functions converge absolutely and uniformly on all compact sets in the $z$-plane. …
##### 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the $z$-plane. …
##### 8: 1.16 Distributions
If the support of $\phi$ is a compact set1.9(vii)), then $\phi$ is called a function of compact support. … A sequence $\{\phi_{n}\}$ of test functions converges to a test function $\phi$ if the support of every $\phi_{n}$ is contained in a fixed compact set $K$ and as $n\to\infty$ the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly on $K$ to $\phi^{(k)}$ for $k=0,1,2,\dots$. … Let $\mathcal{D}({\mathbb{R}}^{n})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in $n$ variables, $\phi(x_{1},x_{2},\dots,x_{n})$, with compact support in ${\mathbb{R}}^{n}$. …
##### 9: 1.10 Functions of a Complex Variable
The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. …
##### 10: 21.7 Riemann Surfaces
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 ${\mathbb{C}}^{2}$, 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 $\tilde{P}$ vanishes. … In this way, we associate a Riemann theta function with every compact Riemann surface $\Gamma$.Then the prime form on the corresponding compact Riemann surface $\Gamma$ is defined by …