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21: 12.19 Tables
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Murzewski and Sowa (1972) includes for , , 7S.
Zhang and Jin (1996, pp. 455–473) includes , , , , and derivatives, , , , 8S; , , and derivatives, , and , , , 8S. Also, first zeros of , , and of derivatives, , 6D; first three zeros of and of derivative, , 6D; first three zeros of and of derivative, , 6D; real and imaginary parts of , , , , , 8S.
22: 19.36 Methods of Computation
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19.36.2
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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►A three-part computational procedure for is described by Franke (1965) for .
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23: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into distinct parts.
denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
… denotes the number of partitions of into odd parts.
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►where the last right-hand side is the sum over of the generating functions for partitions into distinct parts with largest part equal to .
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24: 25.11 Hurwitz Zeta Function
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►As a function of , with () fixed, is analytic in the half-plane .
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►For most purposes it suffices to restrict because of the following straightforward consequences of (25.11.1):
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►Throughout this subsection .
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►As with fixed, ,
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25: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule.
In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.),
IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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26: 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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36.5.7
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►This part of the Stokes set connects two complex saddles.
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►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles.
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27: 14.30 Spherical and Spheroidal Harmonics
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14.30.4
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28: 26.12 Plane Partitions
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►A plane partition, , of a positive integer , is a partition of in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns.
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►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns.
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►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row.
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29: Bibliography S
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory.
American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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30: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Linear operators. Part II.
Wiley Classics Library, John Wiley & Sons, Inc., New York.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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