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21: 10 Bessel Functions
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22: 23 Weierstrass Elliptic and Modular
Functions
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23: 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
24: 3.4 Differentiation
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►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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25: 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.
26: 16.7 Relations to Other Functions
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►For , , symbols see Chapter 34.
Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
27: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
►For large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large. … ►For approximations for the , , and symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.28: 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 ; , ; , .
29: 18.8 Differential Equations
30: Bibliography
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Evaluation of Coulomb wave functions along the transition line.
Physical Rev. (2) 96, pp. 77–79.
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Algorithm 610. A portable FORTRAN subroutine for derivatives of the psi function.
ACM Trans. Math. Software 9 (4), pp. 494–502.
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Theorems on generalized Dedekind sums.
Pacific J. Math. 2 (1), pp. 1–9.
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Numerical Tables for Angular Correlation Computations in -, - and -Spectroscopy: -, -, -Symbols, F- and -Coefficients.
Landolt-Börnstein Numerical Data and Functional Relationships
in Science and Technology, Springer-Verlag.
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A new treatment of the ellipsoidal wave equation.
Proc. London Math. Soc. (3) 9, pp. 21–50.
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