# open point set

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## 1—10 of 63 matching pages

##### 1: 1.6 Vectors and Vector-Valued Functions

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►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

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###### Point Sets in $\u2102$

… ►An*open set*in $\u2102$ is one in which each point has a neighborhood that is contained in the set. … ►A*domain*$D$, say, is an open set in $\u2102$ 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 $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ 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

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${W}_{0}\left(z\right)$ is a single-valued analytic function on $\u2102\setminus (-\mathrm{\infty},-{\mathrm{e}}^{-1}]$, 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 $\u2102\setminus (-\mathrm{\infty},0]$, 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.
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##### 4: 1.5 Calculus of Two or More Variables

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►where $D$ is the image of ${D}^{\ast}$ 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.
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##### 5: 2.3 Integrals of a Real Variable

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►Assume also that ${\partial}^{2}p(\alpha ,t)/{\partial t}^{2}$ 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
$\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial}^{2}p(\alpha ,t)/{\partial t}^{2}$ and $q(\alpha ,t)$ are nonzero.
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##### 6: Mathematical Introduction

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►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).
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$\u2102$ | complex plane (excluding infinity). |
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$(a,b)$ | open interval in $\mathbb{R}$, or open straight-line segment joining $a$ and $b$ in $\u2102$. |

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$(a,b]$ or $[a,b)$ | half-closed intervals. |
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$lim\; inf$ | least limit point. |

$[{a}_{j,k}]$ or $[{a}_{jk}]$ | matrix with $(j,k)$th element ${a}_{j,k}$ or ${a}_{jk}$. |

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##### 7: 15.6 Integral Representations

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►In (15.6.2) the point
$1/z$ lies outside the integration contour, ${t}^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty})$, and ${(1-zt)}^{a}=1$ at $t=0$.
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##### 8: 28.33 Physical Applications

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►As $\omega $ runs from $0$ to $+\mathrm{\infty}$, with $b$ and $f$ fixed, the point
$(q,a)$ moves from $\mathrm{\infty}$ 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.
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##### 9: 1.16 Distributions

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►Let $\varphi $ be a function defined on an open interval $I=(a,b)$, which can be infinite.
The closure of the set of points where $\varphi \ne 0$ is called the

*support*of $\varphi $. If the support of $\varphi $ is a compact set (§1.9(vii)), then $\varphi $ is called a*function of compact support*. … ►The set of tempered distributions is denoted by ${\mathcal{T}}^{\ast}$. … ►Here $\bm{\alpha}$ ranges over a finite set of multi-indices, $P(\mathbf{x})$ is a multivariate polynomial, and $P(\mathbf{D})$ is a partial differential operator. …