About the Project
NIST

argument a fraction

AdvancedHelp

(0.003 seconds)

1—10 of 25 matching pages

1: 15.4 Special Cases
§15.4(iii) Other Arguments
2: Bibliography T
  • I. J. Thompson and A. R. Barnett (1985) COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.
  • I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 159 (3), pp. 241–242.
  • 3: Bibliography C
  • A. R. Curtis (1964b) Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.
  • 4: 35.1 Special Notation
    All fractional or complex powers are principal values.
    a , b complex variables.
    The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( T ) and (of the second kind) B ν ( T ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; T ) or F 1 1 ( a b ; T ) and (of the second kind) Ψ ( a ; b ; T ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; T ) or F 1 2 ( a 1 , a 2 b ; T ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; T ) or F q p ( a 1 , , a p b 1 , , b q ; T ) . An alternative notation for the multivariate gamma function is Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) ) (Herz (1955, p. 480)). Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( T ) = A ν ( T ) / A ν ( 0 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | S , T ) = | T | ν B ν ( S T ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( T ) = | T | ν B ν ( S T ) (Faraut and Korányi (1994, pp. 357–358)).
    5: Bibliography S
  • J. Segura, P. Fernández de Córdoba, and Yu. L. Ratis (1997) A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.
  • 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.
  • K. M. Siegel and F. B. Sleator (1954) Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.
  • K. M. Siegel (1953) An inequality involving Bessel functions of argument nearly equal to their order. Proc. Amer. Math. Soc. 4 (6), pp. 858–859.
  • A. J. Stone and C. P. Wood (1980) Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.
  • 6: 14.32 Methods of Computation
    In particular, for small or moderate values of the parameters μ and ν the power-series expansions of the various hypergeometric function representations given in §§14.3(i)14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …
  • Evaluation (§3.10) of the continued fractions given in §14.14. See Gil and Segura (2000).

  • 7: Bibliography M
  • P. Martín, R. Pérez, and A. L. Guerrero (1992) Two-point quasi-fractional approximations to the Airy function Ai ( x ) . J. Comput. Phys. 99 (2), pp. 337–340.
  • K. S. Miller and B. Ross (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.
  • R. Morris (1979) The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.
  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.
  • C. Mortici (2013a) A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.
  • 8: 33.8 Continued Fractions
    §33.8 Continued Fractions
    With arguments η , ρ suppressed, …
    a = 1 + ± i η ,
    The continued fraction (33.8.1) converges for all finite values of ρ , and (33.8.2) converges for all ρ 0 . … The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. …
    9: 16.4 Argument Unity
    §16.4 Argument Unity
    The function F q q + 1 with argument unity and general values of the parameters is discussed in Bühring (1992). … with limiting form a ( ψ ( a + n + 1 ) - ψ ( a ) ) = a d d a ( a ) n + 1 ( a ) n + 1 in the case that c = a + 1 . …
    §16.4(iv) Continued Fractions
    For continued fractions for ratios of F 2 3 functions with argument unity, see Cuyt et al. (2008, pp. 315–317). …
    10: 7.22 Methods of Computation
    Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. … The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). … For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).