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11—20 of 582 matching pages
11: 3.4 Differentiation
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►The are the differentiated Lagrangian interpolation coefficients:
…where is as in (3.3.10).
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►where is a simple closed contour described in the positive rotational sense such that and its interior lie in the domain of analyticity of , and is interior to .
Taking to be a circle of radius centered at , we obtain
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12: 26.5 Lattice Paths: Catalan Numbers
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is the Catalan number.
…(Sixty-six equivalent definitions of are given in Stanley (1999, pp. 219–229).)
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26.5.3
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26.5.4
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26.5.7
13: 14.33 Tables
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Abramowitz and Stegun (1964, Chapter 8) tabulates for , , 5–8D; for , , 5–7D; and for , , 6–8D; and for , , 6S; and for , , 6S. (Here primes denote derivatives with respect to .)
Zhang and Jin (1996, Chapter 4) tabulates for , , 7D; for , , 8D; for , , 8S; for , , 8D; for , , , , 8S; for , , 8S; for , , , 5D; for , , 7S; for , , 8S. Corresponding values of the derivative of each function are also included, as are 6D values of the first 5 -zeros of and of its derivative for , .
Belousov (1962) tabulates (normalized) for , , , 6D.
14: Bibliography P
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A new basis for the representation of the rotation group. Lamé and Heun polynomials.
J. Mathematical Phys. 14 (8), pp. 1130–1139.
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Bounds for ratios of modified Bessel functions.
Integral Transform. Spec. Funct. 9 (4), pp. 293–298.
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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria.
Comput. J. 9 (4), pp. 404–407.
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Primzahlverteilung.
Die Grundlehren der mathematischen Wissenschaften, Vol. 91, Springer-Verlag, Berlin-Göttingen-Heidelberg (German).
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Stacking models of vesicles and compact clusters.
J. Statist. Phys. 80 (3–4), pp. 755–779.
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15: Bibliography K
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Branch Cuts for Complex Elementary Functions or Much Ado About Nothing’s Sign Bit.
In The State of the Art in Numerical Analysis (Birmingham, 1986), A. Iserles and M. J. D. Powell (Eds.),
Inst. Math. Appl. Conf. Ser. New Ser., Vol. 9, pp. 165–211.
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Determinant structure of the rational solutions for the Painlevé II equation.
J. Math. Phys. 37 (9), pp. 4693–4704.
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Poly-Bernoulli numbers.
J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.
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Overview of some new results concerning the theory and applications of the Rayleigh special function.
Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.
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On the zeros of the Fresnel integrals.
Canad. J. Math. 9, pp. 118–131.
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16: Bibliography L
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Exact computation of the - symbols.
Comput. Phys. Comm. 70 (3), pp. 544–556.
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Statistical Physics, Part 2: Theory of the Condensed State.
Pergamon Press, Oxford.
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Asymptotics of the first Appell function with large parameters.
Integral Transforms Spec. Funct. 24 (9), pp. 715–733.
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Monotonicity of the differences of zeros of Bessel functions as a function of order.
Proc. Amer. Math. Soc. 15 (1), pp. 91–96.
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Deep water ship-waves.
Phil. Mag. 9, pp. 733–757.
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17: 34 3j, 6j, 9j Symbols
Chapter 34 Symbols
…18: 1.12 Continued Fractions
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is called the th approximant or convergent to
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and are called the th (canonical) numerator and denominator respectively.
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►Define
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►Conversely, is called an extension of .
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►Then the convergents satisfy
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19: 1.11 Zeros of Polynomials
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►Set to reduce to , with , .
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, , , .
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►Resolvent cubic is with roots , , , and , , .
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►Let
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►Then , with , is stable iff ; , ; , .
20: Bibliography N
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Generalization of Binet’s Gamma function formulas.
Integral Transforms Spec. Funct. 24 (8), pp. 597–606.
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On the large argument asymptotics of the Lommel function via Stieltjes transforms.
Asymptot. Anal. 91 (3-4), pp. 265–281.
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On the calculation of elliptic integrals of the second and third kinds.
Zastos. Mat. 11, pp. 91–94.
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Der Eulersche Dilogarithmus und seine Verallgemeinerungen.
Nova Acta Leopoldina 90, pp. 123–212.
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The asymptotic behavior of the general real solution of the third Painlevé equation.
Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).
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