# boundary points

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## 1—10 of 24 matching pages

##### 1: 1.9 Calculus of a Complex Variable

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

… ►Any point whose neighborhoods always contain members and nonmembers of $D$ is a*boundary point*of $D$. When its boundary points are added the domain is said to be*closed*, but unless specified otherwise a domain is assumed to be open. … ►A*region*is an open domain together with none, some, or all of its boundary points. Points of a region that are not boundary points are called*interior points*. …##### 2: 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 $V$. …##### 3: 1.13 Differential Equations

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►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda $; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called

*nodes*, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …##### 4: 1.4 Calculus of One Variable

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►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are

*analytic functions*of their variables, and thus ${C}^{\mathrm{\infty}}$, and well defined for all values of these variables; possible exceptions being at boundary points. …##### 5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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►The implicit boundary conditions taken here are that the ${\varphi}_{n}(x)$ and ${\varphi}_{n}^{\prime}(x)$ vanish as $x\to \pm \mathrm{\infty}$, which in this case is equivalent to requiring ${\varphi}_{n}(x)\in {L}^{2}\left(X\right)$, see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point.
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*boundary value*for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{\ast})$ of the form …Boundary values and boundary conditions for the end point $b$ are defined in a similar way. … ►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. … ►See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of $51$ solved Sturm–Liouville problems in Everitt (2005a, pp. 272–331), each with their limit point, or circle, boundary behaviors categorized.##### 6: 3.8 Nonlinear Equations

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►For an arbitrary starting point
${z}_{0}\in \u2102$, convergence cannot be predicted, and the boundary of the set of points
${z}_{0}$ that generate a sequence converging to a particular zero has a very complicated structure.
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##### 7: 20.2 Definitions and Periodic Properties

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►For fixed $z$, each of ${\theta}_{1}\left(z\right|\tau )/\mathrm{sin}z$, ${\theta}_{2}\left(z\right|\tau )/\mathrm{cos}z$, ${\theta}_{3}\left(z\right|\tau )$, and ${\theta}_{4}\left(z\right|\tau )$ is an analytic function of $\tau $ for $\mathrm{\Im}\tau >0$, with a natural boundary
$\mathrm{\Im}\tau =0$, and correspondingly, an analytic function of $q$ for $$ with a natural boundary
$\left|q\right|=1$.
►The four points
$(0,\pi ,\pi +\tau \pi ,\tau \pi )$ are the vertices of the

*fundamental parallelogram*in the $z$-plane; see Figure 20.2.1. The points …are the*lattice points*. The theta functions are quasi-periodic on the lattice: …##### 8: 10.21 Zeros

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►In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points
$\pm 1$ are the boundaries of $\mathbf{K}$, that is, the eye-shaped domain depicted in Figure 10.20.3.
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►In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points
$\pm 1$ is the lower boundary of $\mathbf{K}$.
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##### 9: 22.3 Graphics

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##### 10: 10.20 Uniform Asymptotic Expansions for Large Order

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►Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.
►The equations of the curved boundaries
${D}_{1}{E}_{1}$ and ${D}_{2}{E}_{2}$ in the $\zeta $-plane are given parametrically by
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►The points
${P}_{1},{P}_{2}$ where these curves intersect the imaginary axis are $\pm \mathrm{i}c$, where
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