# open point set

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##### 1: 1.6 Vectors and Vector-Valued Functions
Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. …
##### 2: 1.9 Calculus of a Complex Variable
###### PointSets in $\mathbb{C}$
An open set in $\mathbb{C}$ is one in which each point has a neighborhood that is contained in the set. … A domain $D$, say, is an open set in $\mathbb{C}$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … Conversely, if at a given point $(x,y)$ the partial derivatives $\ifrac{\partial u}{\partial x}$, $\ifrac{\partial u}{\partial y}$, $\ifrac{\partial v}{\partial x}$, and $\ifrac{\partial v}{\partial y}$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. …
##### 3: 4.13 Lambert $W$-Function
$W_{0}\left(z\right)$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_{k}\left(z\right)$ are single-valued analytic functions on $\mathbb{C}\setminus(-\infty,0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …
##### 4: 1.5 Calculus of Two or More Variables
where $D$ is the image of $D^{*}$ under a mapping $(u,v)\to(x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …
##### 5: 2.3 Integrals of a Real Variable
Assume also that $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ and $q(\alpha,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\ifrac{\partial p(\alpha,t)}{\partial t}$ vanishes, but both $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ and $q(\alpha,t)$ are nonzero. …
##### 6: Mathematical Introduction
These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …
 $\mathbb{C}$ complex plane (excluding infinity). … open interval in $\mathbb{R}$, or open straight-line segment joining $a$ and $b$ in $\mathbb{C}$. …
 $(a,b]$ or $[a,b)$ half-closed intervals. … least limit point. matrix with $(j,k)$th element $a_{j,k}$ or $a_{jk}$. …
##### 7: 15.6 Integral Representations
In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-zt)^{a}=1$ at $t=0$. …
##### 8: 28.33 Physical Applications
As $\omega$ runs from $0$ to $+\infty$, with $b$ and $f$ fixed, the point $(q,a)$ moves from $\infty$ to $0$ along the ray $\mathcal{L}$ given by the part of the line $a=(2b/f)q$ that lies in the first quadrant of the $(q,a)$-plane. …
##### 9: 1.16 Distributions
Let $\phi$ be a function defined on an open interval $I=(a,b)$, which can be infinite. The closure of the set of points where $\phi\not=0$ is called the support of $\phi$. If the support of $\phi$ is a compact set1.9(vii)), then $\phi$ is called a function of compact support. … The set of tempered distributions is denoted by $\mathcal{T}^{*}$. … Here $\boldsymbol{{\alpha}}$ ranges over a finite set of multi-indices, $P(\mathbf{x})$ is a multivariate polynomial, and $P(\mathbf{D})$ is a partial differential operator. …
##### 10: 1.4 Calculus of One Variable
If $f(x)$ is continuous at each point $c\in(a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. … where the supremum is over all sets of points $x_{0} in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …