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21: 16.1 Special Notation
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►The main functions treated in this chapter are the generalized hypergeometric function , the Appell (two-variable hypergeometric) functions , , , , and the Meijer -function .
Alternative notations are , , and for the generalized hypergeometric function, , , , , for the Appell functions, and for the Meijer -function.
nonnegative integers. | |
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arbitrary small positive constant. | |
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22: 22.16 Related Functions
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Approximation for Small
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… ►With as in (22.2.1) and , … ►In Equations (22.16.24)–(22.16.26), . … ►(Sometimes in the literature is denoted by .) …23: 10.69 Uniform Asymptotic Expansions for Large Order
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►Let and be the polynomials defined in §10.41(ii), and
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10.69.3
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10.69.5
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►Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with replaced by .
24: 24.20 Tables
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►Abramowitz and Stegun (1964, Chapter 23) includes exact values of , , ; , , , , 20D; , , 18D.
►Wagstaff (1978) gives complete prime factorizations of and for and , respectively.
In Wagstaff (2002) these results are extended to and , respectively, with further complete and partial factorizations listed up to and , respectively.
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25: 1.12 Continued Fractions
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►when , .
…when , .
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►The even part of exists iff , , and up to equivalence is given by
…The odd part of exists iff , , and up to equivalence is given by
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►where is an arbitrary small positive constant.
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26: 12.9 Asymptotic Expansions for Large Variable
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►Throughout this subsection is an arbitrary small positive constant.
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12.9.1
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12.9.2
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12.9.3
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12.9.4
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27: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Coulomb functions for large charges and small velocities.
Phys. Rev. (2) 97 (2), pp. 542–554.
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Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind.
IMA J. Numer. Anal. 12 (4), pp. 519–526.
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28: 10.17 Asymptotic Expansions for Large Argument
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►and let denote an arbitrary small positive constant.
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►Also, , , and for ,
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►Then the remainder associated with the sum does not exceed the first neglected term in absolute value and has the same sign provided that .
Similarly for , provided that .
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►If these expansions are terminated when , then the remainder term is bounded in absolute value by the first neglected term, provided that .
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29: Bibliography L
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Inequalities and approximations for zeros of Bessel functions of small order.
SIAM J. Math. Anal. 14 (2), pp. 383–388.
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New method to obtain small parameter power series expansions of Mathieu radial and angular functions.
Math. Comp. 78 (265), pp. 255–274.
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Note sur la fonction
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Acta Math. 11 (1-4), pp. 19–24 (French).
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