# closed point set

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## 1—10 of 56 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. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …##### 2: 1.9 Calculus of a Complex Variable

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##### 3: 2.1 Definitions and Elementary Properties

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►If the set
$\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta $, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty}$.
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##### 4: 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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##### 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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${f(z)|}_{C}=0$ | $f(z)$ is continuous at all points of a simple closed contour $C$ in $\u2102$. |

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$[a,b]$ | closed interval in $\mathbb{R}$, or closed straight-line segment joining $a$ and $b$ in $\u2102$. |

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

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##### 7: 10.25 Definitions

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►In particular, the

*principal branch*of ${I}_{\nu}\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu}$, is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. … ►
10.25.3
$${K}_{\nu}\left(z\right)\sim \sqrt{\pi /(2z)}{\mathrm{e}}^{-z},$$

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►It has a branch point at $z=0$ for all $\nu \in \u2102$.
The *principal branch*corresponds to the principal value of the square root in (10.25.3), is analytic in $\u2102\setminus (-\mathrm{\infty},0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi $. …##### 8: 1.10 Functions of a Complex Variable

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►If $D=\u2102\setminus (-\mathrm{\infty},0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta}$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with $$ in both cases.
Similarly if $D=\u2102\setminus [0,\mathrm{\infty})$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with $$ in both cases.
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►Alternatively, take ${z}_{0}$ to be any point in $D$ and set
$F({z}_{0})={\mathrm{e}}^{\alpha \mathrm{ln}\left(1-{z}_{0}\right)}{\mathrm{e}}^{\beta \mathrm{ln}\left(1+{z}_{0}\right)}$ where the logarithms assume their principal values.
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►Let $D$ be a domain and $[a,b]$ be a closed finite segment of the real axis.
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##### 9: 1.8 Fourier Series

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►If a function $f(x)\in {C}^{2}[0,2\pi ]$ is periodic, with period $2\pi $, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to ${f}^{\prime}(x)$.
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