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11: 1.2 Elementary Algebra
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►The geometric mean
and harmonic mean
of positive numbers are given by
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1.2.41
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1.2.42
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1.2.43
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1.2.44
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12: Viewing DLMF Interactive 3D Graphics
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►Below we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer.
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13: Foreword
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►The online version, the NIST Digital
Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium.
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14: 3.12 Mathematical Constants
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►For access to online high-precision numerical values of mathematical constants see Sloane (2003).
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15: 1.1 Special Notation
16: Bibliography L
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Lamé-Wangerin functions.
Quart. J. Math., Oxford Ser. (2) 3, pp. 107–114.
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Addendum to: “Changing the order of integration”.
J. Austral. Math. Soc. 14, pp. 383–384.
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Numerical Evaluation of Special Functions.
In Mathematics of Computation 1943–1993: A Half-Century of
Computational Mathematics (Vancouver, BC, 1993),
Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.
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On a set of solutions of the sixth Painlevé equation.
Differ. Uravn. 3 (3), pp. 520–523 (Russian).
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Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray.
Math. Comp. 17 (84), pp. 395–404.
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17: Notices
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Index of Selected Software Within the DLMF Chapters
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Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.
In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.
18: Bibliography N
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An explicit formula for the coefficients in Laplace’s method.
Constr. Approx. 38 (3), pp. 471–487.
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Error bounds and exponential improvement for the asymptotic expansion of the Barnes -function.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.
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Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal.
Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.
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Handbuch der Theorie der Gammafunktion.
B. G. Teubner, Leipzig (German).
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Diffraction Effects in Semiclassical Scattering.
Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.
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19: Bibliography V
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An integral transform involving Heun functions and a related eigenvalue problem.
SIAM J. Math. Anal. 17 (3), pp. 688–703.
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On the zeros of the Riemann zeta function in the critical strip. IV.
Math. Comp. 46 (174), pp. 667–681.
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Integrating products of Bessel functions with an additional exponential or rational factor.
Comput. Phys. Comm. 178 (8), pp. 578–590.
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A Mathieu equation for ships rolling among waves. I, II.
Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.
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Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories.
Il Nuovo Cimento A 57 (1), pp. 190–197.
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20: Bibliography M
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Hypergeometric Functions.
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Algorithm AS 63. The incomplete beta integral.
Appl. Statist. 22 (3), pp. 409–411.
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On the evaluation of the integral over the product of two spherical Bessel functions.
J. Math. Phys. 32 (3), pp. 642–648.
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The exponential integral distribution.
Statist. Probab. Lett. 5 (3), pp. 209–211.
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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation.
SIAM J. Numer. Anal. 3 (3), pp. 390–409.
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