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1: 19.37 Tables
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 7S by Beli͡akov et al. (1962).
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►Tabulated for , to 10D by Fettis and Caslin (1964).
►Tabulated for , to 6D by Byrd and Friedman (1971), for , and to 8D by Abramowitz and Stegun (1964, Chapter 17), and for , to 9D by Zhang and Jin (1996, pp. 674–675).
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►Tabulated for , , to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)).
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2: Bibliography
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Chapter of Ramanujan’s second notebook: Theta-functions and -series.
Mem. Amer. Math. Soc. 53 (315), pp. v+85.
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Hypergeometric Functions and Elliptic Integrals.
In Current Topics in Analytic Function Theory, H. M. Srivastava and S. Owa (Eds.),
pp. 48–85.
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Dirichlet series related to the Riemann zeta function.
J. Number Theory 19 (1), pp. 85–102.
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Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices.
Bulletin of the American Mathematical Society 6 (1), pp. 81–85.
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3: 26.6 Other Lattice Path Numbers
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Delannoy Number
► is the number of paths from to that are composed of directed line segments of the form , , or . ►
26.6.1
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26.6.10
,
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4: 4.48 Software
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►These will not be listed here.
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►Citations in bulleted lists refer to papers for which research software has been made available and can be downloaded via the Web.
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•
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Kearfott (1996). Fortran 90.
5: Bibliography K
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A proof of the -Macdonald-Morris conjecture for
.
Mem. Amer. Math. Soc. 108 (516), pp. vi+80.
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Pascal program for generating tables of Clebsch-Gordan coefficients.
Comput. Phys. Comm. 85 (1), pp. 82–88.
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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type.
ACM Trans. Math. Software 22 (4), pp. 385–392.
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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials.
Appl. Anal. 90 (3-4), pp. 731–746.
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Jacobi Functions and Analysis on Noncompact Semisimple Lie Groups.
In Special Functions: Group Theoretical Aspects and Applications,
pp. 1–85.
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6: 3.4 Differentiation
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►where is as in (3.3.10).
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►If can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
…Taking to be a circle of radius centered at , we obtain
…The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).
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►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.
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7: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…( is defined to be 0.)
…It can be expressed as a sum over all primes :
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►is the sum of the th powers of the divisors of , where the exponent can be real or complex.
Note that .
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8: Joyce E. Conlon
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►From 1980–85 she worked as a computer programmer for the Defense Mapping Agency.
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9: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Integrals that contain a probability function of complicated arguments.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Some properties and applications of the repeated integrals of the error function.
Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.
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10: Bibliography T
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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis.
Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.
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LSFBTR: A subroutine for calculating spherical Bessel transforms.
Comput. Phys. Comm. 30 (1), pp. 93–99.
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Improved error bounds for the Liouville-Green (or WKB) approximation.
J. Math. Anal. Appl. 85 (1), pp. 79–89.
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Modified Bessel functions and of real order and complex argument, to selected accuracy.
Comput. Phys. Comm. 47 (2-3), pp. 245–257.
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Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners.
John Wiley & Sons Inc., New York.
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