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11: Bibliography Y
  • T. Yoshida (1995) Computation of Kummer functions U ( a , b , x ) for large argument x by using the τ -method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).
  • 12: Bibliography S
  • J. Segura and A. Gil (1998) Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.
  • 13: Bibliography Z
  • M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.
  • J. Zhang and J. A. Belward (1997) Chebyshev series approximations for the Bessel function Y n ( z ) of complex argument. Appl. Math. Comput. 88 (2-3), pp. 275–286.
  • S. Zhang and J. Jin (1996) Computation of Special Functions. John Wiley & Sons Inc., New York.
  • 14: Bibliography O
  • I. Olkin (1959) A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.
  • F. W. J. Olver and J. M. Smith (1983) Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.
  • C. Osácar, J. Palacián, and M. Palacios (1995) Numerical evaluation of the dilogarithm of complex argument. Celestial Mech. Dynam. Astronom. 62 (1), pp. 93–98.
  • 15: Bibliography W
  • T. Watanabe, M. Natori, and T. Oguni (Eds.) (1994) Mathematical Software for the P.C. and Work Stations – A Collection of Fortran 77 Programs. North-Holland Publishing Co., Amsterdam.
  • 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.
  • T. Weider (1999) Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.
  • J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.
  • G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.
  • 16: Bibliography F
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.
  • L. Fox (1960) Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments. National Physical Laboratory Mathematical Tables, Vol. 4. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.
  • C. L. Frenzen (1992) Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.
  • T. Fukushima (2010) Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.
  • 17: 10.74 Methods of Computation
    The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument x or z is sufficiently small in absolute value. … Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large x or | z | , whether or not ν is large. …
    18: Errata
  • Equation (35.7.3)

    Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument F 1 2 was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

  • Equations (33.11.2)–(33.11.7)

    The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.

  • Equations (9.7.3), (9.7.4)

    Originally the function χ was presented with argument given by a positive integer n . It has now been clarified to be valid for argument given by a positive real number x .

  • Section 17.1

    The notation used for the q -Appell functions in Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument q , as used in Gasper and Rahman (2004).

  • Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

    These equations were rewritten with the modulus (second argument) of the Jacobian elliptic function defined explicitly in the preceding line of text.

  • 19: 10.18 Modulus and Phase Functions
    §10.18(iii) Asymptotic Expansions for Large Argument
    10.18.20 - ( 2 k - 3 ) !! ( 2 k ) !! ( μ - 1 ) ( μ - 9 ) ( μ - ( 2 k - 3 ) 2 ) ( μ - ( 2 k + 1 ) ( 2 k - 1 ) 2 ) ( 2 x ) 2 k , k 2 ,
    The remainder after k terms in (10.18.17) does not exceed the ( k + 1 ) th term in absolute value and is of the same sign, provided that k > ν - 1 2 .
    20: 19.29 Reduction of General Elliptic Integrals
    Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. …where the arguments of the R D function are, in order, ( a - b ) ( u - c ) , ( b - c ) ( a - u ) , ( a - b ) ( b - c ) . …