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1: Software Index
‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Open Source Collections and Systems.

    These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • Guide to Available Mathematical Software

    A cross index of mathematical software in use at NIST.

  • 2: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • G. Wei and B. E. Eichinger (1993) Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.
  • J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.
  • A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.
  • J. Wimp (1985) Some explicit Padé approximants for the function Φ / Φ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.
  • 3: Bibliography K
  • M. Kodama (2008) Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.
  • P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.
  • K. S. Kölbig (1972c) Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Comput. Phys. Comm. 4, pp. 221–226.
  • T. H. Koornwinder and I. Sprinkhuizen-Kuyper (1978) Hypergeometric functions of 2 × 2 matrix argument are expressible in terms of Appel’s functions F 4 . Proc. Amer. Math. Soc. 70 (1), pp. 39–42.
  • P. Kravanja, O. Ragos, M. N. Vrahatis, and F. A. Zafiropoulos (1998) ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.
  • 4: Bibliography
  • D. E. Amos (1985) A subroutine package for Bessel functions of a complex argument and nonnegative order. Technical Report Technical Report SAND85-1018, Sandia National Laboratories, Albuquerque, NM.
  • D. E. Amos (1986) Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software 12 (3), pp. 265–273.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • R. W. B. Ardill and K. J. M. Moriarty (1978) Spherical Bessel functions j n and y n of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.
  • 5: Bibliography R
  • Yu. L. Ratis and P. Fernández de Córdoba (1993) A code to calculate (high order) Bessel functions based on the continued fractions method. Comput. Phys. Comm. 76 (3), pp. 381–388.
  • F. E. Relton (1965) Applied Bessel Functions. Dover Publications Inc., New York.
  • G. F. Remenets (1973) Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.
  • S. R. Rengarajan and J. E. Lewis (1980) Mathieu functions of integral orders and real arguments. IEEE Trans. Microwave Theory Tech. 28 (3), pp. 276–277.
  • D. St. P. Richards (2004) Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.
  • 6: Bibliography O
  • O. M. Ogreid and P. Osland (1998) Summing one- and two-dimensional series related to the Euler series. J. Comput. Appl. Math. 98 (2), pp. 245–271.
  • S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.
  • S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.
  • A. B. Olde Daalhuis (1994) Asymptotic expansions for q -gamma, q -exponential, and q -Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.
  • I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.
  • 7: Bibliography G
  • W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.
  • A. Gervois and H. Navelet (1984) Some integrals involving three Bessel functions when their arguments satisfy the triangle inequalities. J. Math. Phys. 25 (11), pp. 3350–3356.
  • A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.
  • E. T. Goodwin (1949a) Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.
  • K. I. Gross and D. St. P. Richards (1987) Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Amer. Math. Soc. 301 (2), pp. 781–811.
  • 8: Bibliography M
  • S. Makinouchi (1966) Zeros of Bessel functions J ν ( x ) and Y ν ( x ) accurate to twenty-nine significant digits. Technology Reports of the Osaka University 16 (685), pp. 1–44.
  • J. McMahon (1894) On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • R. Morris (1979) The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.
  • M. E. Muldoon (1981) The variation with respect to order of zeros of Bessel functions. Rend. Sem. Mat. Univ. Politec. Torino 39 (2), pp. 15–25.
  • 9: Bibliography C
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n α ( x )  as the index α  and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
  • J. B. Campbell (1979) Bessel functions J ν ( x ) and Y ν ( x ) of real order and real argument. Comput. Phys. Comm. 18 (1), pp. 133–142.
  • J. B. Campbell (1981) Bessel functions I ν ( x ) and K ν ( x ) of real order and complex argument. Comput. Phys. Comm. 24 (1), pp. 97–105.
  • W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.
  • J. P. Coleman (1980) A Fortran subroutine for the Bessel function J n ( x ) of order 0 to 10 . Comput. Phys. Comm. 21 (1), pp. 109–118.