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11: Bille C. Carlson
This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. …
12: Notices
  • Index of Selected Software Within the DLMF Chapters

    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.

  • 13: 31.14 General Fuchsian Equation
    The three sets of parameters comprise the singularity parameters a j , the exponent parameters α , β , γ j , and the N 2 free accessory parameters q j . With a 1 = 0 and a 2 = 1 the total number of free parameters is 3 N 3 . …
    14: Bibliography N
  • M. Neher (2007) Complex standard functions and their implementation in the CoStLy library. ACM Trans. Math. Softw. 33 (1), pp. Article 2.
  • NetNUMPAC (free Fortran library)
  • NMS (free collection of Fortran subroutines)
  • 15: 5.20 Physical Applications
    Suppose the potential energy of a gas of n point charges with positions x 1 , x 2 , , x n and free to move on the infinite line < x < , is given by … For n charges free to move on a circular wire of radius 1 , …
    16: 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. …
    17: 16.10 Expansions in Series of F q p Functions
    Here α , β , and γ are free real or complex parameters. …
    18: 21.9 Integrable Equations
    These parameters, including 𝛀 , are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition u ( x , y , 0 ) (Deconinck and Segur (1998)). …
    19: Bibliography F
  • FDLIBM (free C library)
  • FN (free Fortran library)
  • L. W. Fullerton (1977) Portable Special Function Routines. In Portability of Numerical Software (Oak Brook, Illinois, 1976), W. R. Cowell (Ed.), Lecture Notes in Computer Science, Vol. 57, pp. 452–483.
  • 20: 18.21 Hahn Class: Interrelations
    See accompanying text
    Figure 18.21.1: Askey scheme. The number of free real parameters is zero for Hermite polynomials. It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. … Magnify