boundary points
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1: 1.9 Calculus of a Complex Variable
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Point Sets in
… ►Any point whose neighborhoods always contain members and nonmembers of is a boundary point of . 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 plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii).
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►and be the closed and bounded point set in the plane having a simple closed curve as boundary.
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►Suppose is a piecewise smooth surface which forms the complete boundary of a bounded closed point set , and is oriented by its normal being outwards from .
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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, ; (ii) the corresponding (real) eigenfunctions, and , have the same number of zeros, also called nodes, for as for ; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points.
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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 , and well defined for all values of these variables; possible exceptions being at boundary points.
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5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►The implicit boundary conditions taken here are that the and vanish as , which in this case is equivalent to requiring , see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point.
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► A boundary value for the end point
is a linear form on of the form
…Boundary values and boundary conditions for the end point
are defined in a similar way.
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►The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications.
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►See, in particular, the overview Everitt (2005b, pp. 45–74), and the uniformly annotated listing of 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
, convergence cannot be predicted, and the boundary of the set of points
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 , each of , , , and is an analytic function of for , with a natural boundary
, and correspondingly, an analytic function of for with a natural boundary
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►The four points
are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1.
The points
…are the lattice points.
The theta functions are quasi-periodic on the lattice:
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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
are the boundaries of , 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
is the lower boundary of .
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9: 22.3 Graphics
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