About the Project

special distributions


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11: 1.17 Integral and Series Representations of the Dirac Delta
In applications in physics and engineering, the Dirac delta distribution1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) δ ( x ) . … (1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). …
12: DLMF Project News
error generating summary
13: Bibliography K
  • E. G. Kalnins, W. Miller, G. F. Torres del Castillo, and G. C. Williams (2000) Special Functions and Perturbations of Black Holes. In Special Functions (Hong Kong, 1999), pp. 140–151.
  • A. Ya. Kazakov and S. Yu. Slavyanov (1996) Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.
  • N. D. Kazarinoff (1988) Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.
  • M. K. Kerimov (2008) Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.
  • S. H. Khamis (1965) Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).
  • 14: 7.20 Mathematical Applications
    The spiral has several special properties (see Temme (1996b, p. 184)). … The normal distribution function with mean m and standard deviation σ is given by …For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).
    15: Bibliography I
  • IMSL (commercial C, Fortran, and Java libraries) Visual Numerics, Inc..
  • A. E. Ingham (1932) The Distribution of Prime Numbers. Cambridge Tracts in Mathematics and Mathematical Physics, No. 30, Cambridge University Press, Cambridge.
  • M. E. H. Ismail and E. Koelink (Eds.) (2005) Theory and Applications of Special Functions. Developments in Mathematics, Vol. 13, Springer, New York.
  • M. E. H. Ismail, D. R. Masson, and M. Rahman (Eds.) (1997) Special Functions, q -Series and Related Topics. Fields Institute Communications, Vol. 14, American Mathematical Society, Providence, RI.
  • K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida (1991) From Gauss to Painlevé: A Modern Theory of Special Functions. Aspects of Mathematics E, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, Germany.
  • 16: Notices
    Limited copying and internal distribution of the content of these pages is permitted for research and teaching. Reproduction, copying, or distribution for any commercial purpose is strictly prohibited. … The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. Using special functions in applications often requires computing them. …
  • Master Software Index

    In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

    Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

  • 17: 16.23 Mathematical Applications
    In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three F 2 2 functions are applied to compute the expected value of the excess. … Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. …
    18: 14.30 Spherical and Spheroidal Harmonics
    Special Values
    Distributional Completeness
    The special class of spherical harmonics Y l , m ( θ , ϕ ) , defined by (14.30.1), appear in many physical applications. …
    19: Preface
    Boisvert and Clark were responsible for advising and assisting in matters related to the use of information technology and applications of special functions in the physical sciences (and elsewhere); they also participated in the resolution of major administrative problems when they arose. … Miller was responsible for information architecture, specializing LaTeX for the needs of the project, translation from LaTeX to MathML, and the search interface. … The project was funded in part by NSF Award 9980036, administered by the NSF’s Knowledge and Distributed Intelligence Program. …
    20: Bibliography P
  • J. K. Patel and C. B. Read (1982) Handbook of the Normal Distribution. Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.
  • P. C. B. Phillips (1986) The exact distribution of the Wald statistic. Econometrica 54 (4), pp. 881–895.
  • R. Piessens and M. Branders (1985) A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986b) Integrals and Series: Special Functions, Vol. 2. Gordon & Breach Science Publishers, New York.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1990) Integrals and Series: More Special Functions, Vol. 3. Gordon and Breach Science Publishers, New York.