Clenshaw–Curtis formula (extended)
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31—40 of 289 matching pages
31: 33.14 Definitions and Basic Properties
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►An alternative formula for is
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32: 14.21 Definitions and Basic Properties
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§14.21(iii) Properties
►Many of the properties stated in preceding sections extend immediately from the -interval to the cut -plane . This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …33: 15.11 Riemann’s Differential Equation
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►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1).
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§15.11(ii) Transformation Formulas
►A conformal mapping of the extended complex plane onto itself has the form …34: 27.13 Functions
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►A general formula states that
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►Explicit formulas for have been obtained by similar methods for , and , but they are more complicated.
Exact formulas for have also been found for , and , and for all even .
…Also, Milne (1996, 2002) announce new infinite families of explicit formulas extending Jacobi’s identities.
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35: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Construction of Gauss-Christoffel quadrature formulas.
Math. Comp. 22, pp. 251–270.
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On mean convergence of extended Lagrange interpolation.
J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
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An extended class of orthogonal polynomials defined by a Sturm-Liouville problem.
J. Math. Anal. Appl. 359 (1), pp. 352–367.
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Explicit formulas for Bernoulli numbers.
Amer. Math. Monthly 79, pp. 44–51.
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36: Bibliography N
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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes.
J. Comput. Appl. Math. 39 (2), pp. 193–200.
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On simplified asymptotic formulas for a class of Mathieu functions.
SIAM J. Math. Anal. 15 (6), pp. 1205–1213.
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On a simplified asymptotic formula for the Mathieu function of the third kind.
SIAM J. Math. Anal. 18 (6), pp. 1616–1629.
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An explicit formula for the coefficients in Laplace’s method.
Constr. Approx. 38 (3), pp. 471–487.
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Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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37: 33.21 Asymptotic Approximations for Large
38: 3.4 Differentiation
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Two-Point Formula
… ►Three-Point Formula
… ►Four-Point Formula
… ►Five-Point Formula
… ►If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …39: 8.11 Asymptotic Approximations and Expansions
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8.11.6
, .
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8.11.7
, .
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►This reference also contains explicit formulas for in terms of Stirling numbers and for the case an asymptotic expansion for as .
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►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers.
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