boundary points
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11: 3.7 Ordinary Differential Equations
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►Assume that we wish to integrate (3.7.1) along a finite path from to in a domain .
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§3.7(iii) Taylor-Series Method: Boundary-Value Problems
… ►The remaining two equations are supplied by boundary conditions of the form … ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. … ►General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …12: 1.10 Functions of a Complex Variable
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►If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles.
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►(Or more generally, a simple contour that starts at the center and terminates on the boundary.)
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►Then is a branch point of .
For example, is a branch point of .
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13: 10.41 Asymptotic Expansions for Large Order
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►Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the -plane and the -plane.
The curve in the -plane is the upper boundary of the domain depicted in Figure 10.20.3 and rotated through an angle .
Thus is the point
, where is given by (10.20.18).
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14: 32.2 Differential Equations
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►In general the singularities of the solutions are movable in the sense that their location depends on the constants of integration associated with the initial or boundary conditions.
An equation is said to have the Painlevé property if all its solutions are free from movable branch points; the solutions may have movable poles or movable isolated essential singularities (§1.10(iii)), however.
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15: 18.39 Applications in the Physical Sciences
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►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being and forming a complete set.
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►The solutions of (18.39.8) are subject to boundary conditions at and .
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►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with eigenfunctions vanishing at the end points, in this case see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem.
Namely the th eigenfunction, listed in order of increasing eigenvalues, starting at , has precisely nodes, as real zeros of wave-functions away from boundaries are often referred to.
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►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge.
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16: 23.20 Mathematical Applications
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►For each pair of edges there is a unique point
such that .
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►Let denote the set of points on that are of finite order (that is, those points
for which there exists a positive integer with ), and let be the sets of points with integer and rational coordinates, respectively.
…The resulting points are then tested for finite order as follows.
…If any of these quantities is zero, then the point has finite order.
If any of , , is not an integer, then the point has infinite order.
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17: 28.33 Physical Applications
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§28.33(ii) Boundary-Value Problems
… ►The boundary conditions for (outer clamp) and (inner clamp) yield the following equation for : … ►For a visualization see Gutiérrez-Vega et al. (2003), and for references to other boundary-value problems see: … ►For points that are at intersections of with the characteristic curves or , a periodic solution is possible. …18: 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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►Lastly, users may notice some lack of smoothness in the color boundaries of some of the 4D-type surfaces; see, for example, Figure 10.3.9.
This nonsmoothness arises because the mesh that was used to generate the figure was optimized only for smoothness of the surface, and not for smoothness of the color boundaries.
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complex plane (excluding infinity). | |
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is continuous at all points of a simple closed contour in . | |
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or | half-closed intervals. |
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least limit point. | |
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19: Bibliography S
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The zeros of special functions from a fixed point method.
SIAM J. Numer. Anal. 40 (1), pp. 114–133.
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On the double points of a Mathieu equation.
J. Comput. Appl. Math. 107 (1), pp. 111–125.
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Rational Points on Elliptic Curves.
Undergraduate Texts in Mathematics, Springer-Verlag, New York.
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Some Boundary Value Problems Associated with the Heun Equation.
Ph.D. Thesis, London University.
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Mixed Boundary Value Problems in Potential Theory.
North-Holland Publishing Co., Amsterdam.
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20: Bibliography
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Normal forms of functions near degenerate critical points, the Weyl groups and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Normal forms of functions in the neighborhood of degenerate critical points.
Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).
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Critical points of smooth functions, and their normal forms.
Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).
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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.
Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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