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11: 18.40 Methods of Computation
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►In what follows we consider only the simple, illustrative, case that is continuously differentiable so that , with real, positive, and continuous on a real interval The strategy will be to: 1) use the moments to determine the recursion coefficients of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas and weights (or Christoffel numbers) from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32).
These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to , as will be considered in the following paragraphs.
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►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let be a positive integer and define
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Histogram Approach
… ►In what follows this is accomplished in two ways: i) via the Lagrange interpolation of §3.3(i) ; and ii) by constructing a pointwise continued fraction, or PWCF, as follows: …12: 31.8 Solutions via Quadratures
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►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows.
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►The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).
13: 7.20 Mathematical Applications
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►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv).
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14: 25.3 Graphics
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15: Philip J. Davis
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►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined.
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16: 22.8 Addition Theorems
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►Greenhill (1959, pp. 121–130) reviews these results in terms of the geometric poristic polygon constructions of Poncelet.
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17: 20.11 Generalizations and Analogs
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►Similar identities can be constructed for , , and .
…For applications to rapidly convergent expansions for see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004).
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18: Bibliography G
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Construction of Gauss-Christoffel quadrature formulas.
Math. Comp. 22, pp. 251–270.
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New method for constructing wavefunctions for bound states and scattering.
J. Chem. Phys. 51, pp. 14–25.
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Constructing wavefunctions for nonlocal potentials.
J. Chem. Phys. 52, pp. 6211–6217.
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19: Preface
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►The DLMF has been constructed specifically for effective Web usage and contains features unique to Web presentation.
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20: 4.7 Derivatives and Differential Equations
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►When is a general power, is replaced by the branch of used in constructing
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