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►Then is an isolatedsingularity of .
…Lastly, if for infinitely many negative , then is an isolatedessentialsingularity.
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►In any neighborhood of an isolatedessentialsingularity, however small, an analytic function assumes every value in with at most one exception.
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►be a nonlinear second-order differential equation in which is a rational function of and , and is locally analytic in , that is, analytic except for isolatedsingularities in .
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 isolatedessentialsingularities (§1.10(iii)), however.
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►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where is continuous, with convergence to if is an isolated point of discontinuity.
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►An essential feature of such symmetric operators is that their eigenvalues are real, and eigenfunctions
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►In unusual cases , even for all , such as in the case of the Schrödinger–Coulomb problem () discussed in §18.39 and §33.14, where the point spectrum actually accumulates at the onset of the continuum at , implying an essentialsingularity, as well as a branch point, in matrix elements of the resolvent, (1.18.66).
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►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
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►Partial sums of Fourier orthogonal polynomial expansions are polynomials of best approximation in space and they are also the essential building blocks for approximation in spaces.
…OPs are essential for developing approximation theory on regular domains, including characterization of best approximation.
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►Fourier orthogonal expansions provide essential tools and building blocks for harmonic analysis and computational harmonic analysis.
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►The common zeros of , if they exist, are all real, simple, and isolated points.
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►There are essentially five admissible PDOs with nonnegative weight function:
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►►►Figure 18.39.2: Coulomb–Pollaczek weight functions, , (18.39.50) for , , and .
For the weight function, red curve, has an essentialsingularity at , as all derivatives vanish as ; the green curve is , to be compared with its histogram approximation in §18.40(ii).
For the weight function, blue curve, is non-zero at , but this point is also an essentialsingularity as the discrete parts of the weight function of (18.39.51) accumulate as , .
Magnify
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►The Schrödinger operator essentialsingularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as .
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►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter.
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►However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential.
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►The problems sketched in §§2.3(v) and 2.4(v) involve only two of many possibilities for the coalescence of endpoints, saddle points, and singularities in integrals associated with the special functions.
…For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions.
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►For two coalescing saddle points and an algebraic singularity see Temme (1986), Jin and Wong (1998).
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