About the Project

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11: About the Project
The results have been published in book form as the NIST Handbook of Mathematical Functions, by Cambridge University Press, and disseminated in the free electronic Digital Library of Mathematical Functions. …Details of the early history of the DLMF Project are given in the Preface and on pp.  ix–xi in the NIST Handbook of Mathematical Functions. … After the death in April 2013 of Frank W. … They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. …
12: 22.20 Methods of Computation
A powerful way of computing the twelve Jacobian elliptic functions for real or complex values of both the argument z and the modulus k is to use the definitions in terms of theta functions given in §22.2, obtaining the theta functions via methods described in §20.14. … By application of the transformations given in §§22.7(i) and 22.7(ii), k or k can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. … From the first two terms in (22.10.6) we find dn ( 0.19 , 1 19 ) = 0.999951 . Then by using (22.7.4) we have dn ( 0.2 , 0.19 ) = 0.996253 . … Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …
13: 20.3 Graphics
See accompanying text
Figure 20.3.10: θ 1 ( π x , q ) , 0 x 2 , 0 q 0.99 . Magnify 3D Help
See accompanying text
Figure 20.3.11: θ 2 ( π x , q ) , 0 x 2 , 0 q 0.99 . Magnify 3D Help
See accompanying text
Figure 20.3.12: θ 3 ( π x , q ) , 0 x 2 , 0 q 0.99 . Magnify 3D Help
In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. … In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. …
14: 32.3 Graphics
See accompanying text
Figure 32.3.3: w k ( x ) for 12 x 0.73 and k = 1.85185 3 , 1.85185 5 . …The parabola 6 w 2 + x = 0 is shown in black. Magnify
See accompanying text
Figure 32.3.4: w k ( x ) for 12 x 2.3 and k = 0.45142 7 , 0.45142 8 . …The parabola 6 w 2 + x = 0 is shown in black. Magnify
See accompanying text
Figure 32.3.6: w k ( x ) for 10 x 4 with k = 0.999 , 1.001 . …The parabola 2 w 2 + x = 0 is shown in black. Magnify
If we set d 2 u / d x 2 = 0 in (32.3.2) and solve for u , then …
15: 9.9 Zeros
They are denoted by a k , a k , b k , b k , respectively, arranged in ascending order of absolute value for k = 1 , 2 , . They lie in the sectors 1 3 π < ph z < 1 2 π and 1 2 π < ph z < 1 3 π , and are denoted by β k , β k , respectively, in the former sector, and by β k ¯ , β k ¯ , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for k = 1 , 2 , . See §9.3(ii) for visualizations. For the distribution in of the zeros of Ai ( z ) σ Ai ( z ) , where σ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … For error bounds for the asymptotic expansions of a k , b k , a k , and b k see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). … Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of Bi and Bi in the upper half plane. …
16: 23.4 Graphics
(The figures in this subsection may be compared with the figures in §22.3(i).) … (The figures in this subsection may be compared with the figures in §22.3(iii).) …
17: Bibliography L
  • A. Laforgia (1979) On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.
  • E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.
  • D. H. Lehmer (1941) Guide to Tables in the Theory of Numbers. Bulletin of the National Research Council, No. 105, National Research Council, Washington, D.C..
  • W. J. Lentz (1976) Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.
  • D. W. Lozier and F. W. J. Olver (1994) 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.
  • 18: 22.19 Physical Applications
    With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is … Classical motion in one dimension is described by Newton’s equation … Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. … The classical rotation of rigid bodies in free space or about a fixed point may be described in terms of elliptic, or hyperelliptic, functions if the motion is integrable (Audin (1999, Chapter 1)). …Elementary discussions of this topic appear in Lawden (1989, §5.7), Greenhill (1959, pp. 101–103), and Whittaker (1964, Chapter VI). …
    19: DLMF Project News
    error generating summary
    20: Bibliography N
  • A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.
  • NetNUMPAC (free Fortran library)
  • H. M. Nussenzveig (1992) Diffraction Effects in Semiclassical Scattering. Montroll Memorial Lecture Series in Mathematical Physics, Cambridge University Press.
  • J. F. Nye (2006) Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.
  • J. F. Nye (2007) Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.