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21: 8.23 Statistical Applications
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►In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of ; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).
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22: 26.6 Other Lattice Path Numbers
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23: 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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24: 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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25: 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.
26: 22.7 Landen Transformations
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27: 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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28: 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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29: 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. …30: 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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