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1: Bibliography D
  • A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • T. M. Dunster (1999) Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.
  • T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.
  • 2: Bibliography N
  • G. Nemes (2017a) Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.
  • G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.
  • G. Nemes (2014a) Error bounds and exponential improvement for the asymptotic expansion of the Barnes G -function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.
  • G. Nemes (2015a) Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 3: Bibliography O
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
  • F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
  • 4: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • 5: Bibliography B
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • M. V. Berry and C. J. Howls (1994) Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.
  • J. M. Blair, C. A. Edwards, and J. H. Johnson (1976) Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.
  • S. Bochner (1952) Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
  • 6: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • I. A. Stegun and R. Zucker (1981) Automatic computing methods for special functions. IV. Complex error function, Fresnel integrals, and other related functions. J. Res. Nat. Bur. Standards 86 (6), pp. 661–686.
  • A. Strecok (1968) On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.
  • 7: Bibliography F
  • FDLIBM (free C library)
  • H. E. Fettis, J. C. Caslin, and K. R. Cramer (1973) Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.
  • C. L. Frenzen (1987a) Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.
  • C. L. Frenzen (1990) Error bounds for a uniform asymptotic expansion of the Legendre function Q n m ( cosh z ) . SIAM J. Math. Anal. 21 (2), pp. 523–535.
  • 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.
  • 8: Software Index
    Open Source With Book Commercial
    20 Theta Functions
    ‘✓’ 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. …
  • 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.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    9: 28.8 Asymptotic Expansions for Large q
    For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). For error estimates see Kurz (1979), and for graphical interpretation see Figure 28.2.1. … For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … They are derived by rigorous analysis and accompanied by strict and realistic error bounds. … For related results see Langer (1934) and Sharples (1967, 1971). …
    10: Bibliography M
  • A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.
  • H. Majima, K. Matsumoto, and N. Takayama (2000) Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.
  • F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.