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1: Viewing DLMF Interactive 3D Graphics
Viewing DLMF Interactive 3D Graphics
Below we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer. … If you prefer to use a web browser that is not WebGL-enabled, we currently also provide VRML and X3D versions of the DLMF visualizations that may be viewed with a special VRML/X3D browser or plugin. …” VRML (Virtual Reality Modeling Language) is a standard file format for viewing 3D graphics on the web and X3D is its successor. … Linux users must view the visualizations in a WebGL-enabled browser. …
2: 12.15 Generalized Parabolic Cylinder Functions
can be viewed as a generalization of (12.2.4). …See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).
3: DLMF Project News
error generating summary
4: Bibliography T
  • A. Takemura (1984) Zonal Polynomials. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA.
  • J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.
  • N. M. Temme and J. L. López (2001) The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.
  • A. Tucker (2006) Applied Combinatorics. 5th edition, John Wiley and Sons, New York.
  • S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.
  • 5: 18.22 Hahn Class: Recurrence Relations and Differences
    These polynomials satisfy (18.22.2) with p n ( x ) , A n , and C n as in Table 18.22.1. …
    A ~ n = ( n + 2 ( a + b ) 1 ) ( n + a + a ¯ ) ( n + a + b ¯ ) ( 2 n + 2 ( a + b ) 1 ) ( 2 n + 2 ( a + b ) ) ,
    A ( x ) = ( x + α + 1 ) ( x N ) ,
    For A ( x ) , C ( x ) , and λ n in (18.22.12) see Table 18.22.2. …
    C ( x ) = ( x i a ) ( x i b ) .
    6: Need Help?
  • Graphics
  • 7: 9.19 Approximations
    These expansions are for real arguments x and are supplied in sets of four for each function, corresponding to intervals < x a , a x 0 , 0 x b , b x < . The constants a and b are chosen numerically, with a view to equalizing the effort required for summing the series. …
  • Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of Ai ( z ) , Ai ( z ) stored at the nodes. Ai ( z ) and Ai ( z ) are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of Ai ( z ) , Ai ( z ) at the node. Similarly for Bi ( z ) , Bi ( z ) .

  • 8: Brian Antonishek
    Before coming to NIST in 1997, he developed user interaction and scientific visualization techniques for viewing and analyzing 3D engineering data. …
    9: How to Cite
    When citing DLMF from a formal publication, we suggest a format similar to the following:
  • [DLMF]

    NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

  • If you mention a specific equation (or chapter, section, …), you’ll help your readers find it by using our Permalinks & Reference numbers. For example, the sentence in the  source: … The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …
    10: Bibliography W
  • J. Walker (1989) A drop of water becomes a gateway into the world of catastrophe optics. Scientific American 261, pp. 120–123.
  • R. L. Wiegel (1960) A presentation of cnoidal wave theory for practical application. J. Fluid Mech. 7 (2), pp. 273–286.
  • H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“ q ” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.
  • J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.
  • J. Wimp (1965) On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.