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1: 1.9 Calculus of a Complex Variable
Point Sets in
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
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
For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, λ ; (ii) the corresponding (real) eigenfunctions, u ( x ) and w ( t ) , have the same number of zeros, also called nodes, for t ( 0 , c ) as for x ( 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
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 , 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
The implicit boundary conditions taken here are that the ϕ n ( x ) and ϕ n ( x ) vanish as x ± , which in this case is equivalent to requiring ϕ n ( x ) L 2 ( X ) , see Pauling and Wilson (1985, pp. 67–82) for a discussion of this latter point. … A boundary value for the end point a is a linear form on 𝒟 ( ) 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
For an arbitrary starting point z 0 , 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. …
7: 20.2 Definitions and Periodic Properties
For fixed z , each of θ 1 ( z | τ ) / sin z , θ 2 ( z | τ ) / cos z , θ 3 ( z | τ ) , and θ 4 ( z | τ ) is an analytic function of τ for τ > 0 , with a natural boundary τ = 0 , and correspondingly, an analytic function of q for | q | < 1 with a natural boundary | q | = 1 . The four points ( 0 , π , π + τ π , τ π ) 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
In Figures 10.21.1, 10.21.3, and 10.21.5 the two continuous curves that join the points ± 1 are the boundaries of 𝐊 , that is, the eye-shaped domain depicted in Figure 10.20.3. … In Figures 10.21.2, 10.21.4, and 10.21.6 the continuous curve that joins the points ± 1 is the lower boundary of 𝐊 . …
9: 22.3 Graphics
See accompanying text
Figure 22.3.2: k = 0.7 , 3 K x 3 K , K = 1.8456 . For cn ( x , k ) the curve for k = 1 / 2 = 0.70710 is a boundary between the curves that have an inflection point in the interval 0 x 2 K ( k ) , and its translates, and those that do not; see Walker (1996, p. 146). Magnify
10: 10.20 Uniform Asymptotic Expansions for Large Order
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 ζ -plane are given parametrically by … The points P 1 , P 2 where these curves intersect the imaginary axis are ± i c , where …
See accompanying text
Figure 10.20.1: z -plane. P 1 and P 2 are the points ± i c . … Magnify
See accompanying text
Figure 10.20.2: ζ -plane. E 1 and E 2 are the points e π i / 3 ( 3 π / 2 ) 2 / 3 . Magnify