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21: Bibliography D
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Pólya’s Theory of Counting.
In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.),
pp. 144–184.
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Irreducibility of certain generalized Bernoulli polynomials belonging to quadratic residue class characters.
J. Number Theory 25 (1), pp. 72–80.
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On Vandermonde’s theorem, and some more general expansions.
Proc. Edinburgh Math. Soc. 25, pp. 114–132.
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Theta functions and non-linear equations.
Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).
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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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22: 12.14 The Function
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►Other expansions, involving and , can be obtained from (12.4.3) to (12.4.6) by replacing by and by ; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16).
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►Then as
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►Here is as in §12.10(ii), is defined by
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►uniformly for , with , , , and as in §12.10(vii).
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►and and the coefficients and as in §12.10(v).
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23: Publications
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D. W. Lozier, B. R. Miller and B. V. Saunders (1999)
Design of a Digital Mathematical Library for Science, Technology and Education,
Proceedings of the
IEEE Forum on Research and Technology Advances in Digital Libraries (IEEE ADL ’99,
Baltimore, Maryland, May 19, 1999).
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Q. Wang, B. V. Saunders and S. Ressler (2007)
Dissemination of 3D Visualizations of Complex Function Data
for the NIST Digital Library of Mathematical Functions,
CODATA Data Science Journal 6 (2007), pp. S146–S154.
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B. I. Schneider, B. R. Miller and B. V. Saunders (2018)
NIST’s Digital Library of Mathematial Functions,
Physics Today
71, 2, 48 (2018), pp. 48–53.
24: 22.7 Landen Transformations
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25: 36.5 Stokes Sets
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►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
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36.5.4
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►When the Stokes set is given by
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36.5.17
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26: Bibliography P
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The asymptotic behaviour of Pearcey’s integral for complex variables.
Proc. Roy. Soc. London Ser. A 432 (1886), pp. 391–426.
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Fourier Series and Integral Transforms.
Cambridge University Press, Cambridge.
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Voronoi type congruences for Bernoulli numbers.
In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),
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Stacking models of vesicles and compact clusters.
J. Statist. Phys. 80 (3–4), pp. 755–779.
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Elliptic integrals.
Computers in Physics 4 (1), pp. 92–96.
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27: 2.3 Integrals of a Real Variable
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§2.3(ii) Watson’s Lemma
… ►§2.3(iii) Laplace’s Method
… ►where the coefficients are defined by the expansion … ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …28: 34.13 Methods of Computation
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►Methods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
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29: 9.18 Tables
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Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
Yakovleva (1969) tabulates Fock’s functions , , , for . Precision is 7S.