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1: 20 Theta Functions
Chapter 20 Theta Functions
2: 6.20 Approximations
  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • §6.20(ii) Expansions in Chebyshev Series
    §6.20(iii) Padé-Type and Rational Expansions
    3: 6.16 Mathematical Applications
    The n th partial sum is given by …uniformly for x [ π , π ] . … Compare Figure 6.16.1. … It occurs with Fourier-series expansions of all piecewise continuous functions. … …
    4: 7.24 Approximations
  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • §7.24(ii) Expansions in Chebyshev Series
  • Schonfelder (1978) gives coefficients of Chebyshev expansions for x 1 erf x on 0 x 2 , for x e x 2 erfc x on [ 2 , ) , and for e x 2 erfc x on [ 0 , ) (30D).

  • §7.24(iii) Padé-Type Expansions
    5: 10.75 Tables
  • Olver (1960) tabulates j n , m , J n ( j n , m ) , j n , m , J n ( j n , m ) , y n , m , Y n ( y n , m ) , y n , m , Y n ( y n , m ) , n = 0 ( 1 2 ) 20 1 2 , m = 1 ( 1 ) 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n ; see §10.21(viii), and more fully Olver (1954).

  • Bickley et al. (1952) tabulates x n I n ( x ) or e x I n ( x ) , x n K n ( x ) or e x K n ( x ) , n = 2 ( 1 ) 20 , x = 0 (.01 or .1) 10(.1) 20, 8S; I n ( x ) , K n ( x ) , n = 0 ( 1 ) 20 , x = 0 or 0.1 ( .1 ) 20 , 10S.

  • Kerimov and Skorokhodov (1984c) tabulates all zeros of I n 1 2 ( z ) and I n 1 2 ( z ) in the sector 0 ph z 1 2 π for n = 1 ( 1 ) 20 , 9S.

  • The main tables in Abramowitz and Stegun (1964, Chapter 10) give 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 8 , x = 0 ( .1 ) 10 , 5–8S; 𝗃 n ( x ) , 𝗒 n ( x ) n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 , 1 , 2 , x = 0 ( .1 ) 5 , 4–9D; 𝗂 n ( 1 ) ( x ) , 𝗄 n ( x ) , n = 0 ( 1 ) 20 ( 10 ) 50 , 100, x = 1 , 2 , 5 , 10 , 50 , 100 , 10S. (For the notation see §10.1 and §10.47(ii).)

  • Olver (1960) tabulates a n , m , 𝗃 n ( a n , m ) , b n , m , 𝗒 n ( b n , m ) , n = 1 ( 1 ) 20 , m = 1 ( 1 ) 50 , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as n .

  • 6: 28.16 Asymptotic Expansions for Large q
    §28.16 Asymptotic Expansions for Large q
    28.16.1 λ ν ( h 2 ) 2 h 2 + 2 s h 1 8 ( s 2 + 1 ) 1 2 7 h ( s 3 + 3 s ) 1 2 12 h 2 ( 5 s 4 + 34 s 2 + 9 ) 1 2 17 h 3 ( 33 s 5 + 410 s 3 + 405 s ) 1 2 20 h 4 ( 63 s 6 + 1260 s 4 + 2943 s 2 + 486 ) 1 2 25 h 5 ( 527 s 7 + 15617 s 5 + 69001 s 3 + 41607 s ) + .
    7: Bibliography K
  • D. Karp, A. Savenkova, and S. M. Sitnik (2007) Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q -binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
  • 8: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ ( s ) for 0 s 1 (15D), ζ ( s + 1 ) for 0 s 1 (20D), and ln ξ ( 1 2 + i x ) 25.4) for 1 x 1 (20D). For errata see Piessens and Branders (1972).

  • 9: 16.22 Asymptotic Expansions
    §16.22 Asymptotic Expansions
    Asymptotic expansions of G p , q m , n ( z ; 𝐚 ; 𝐛 ) for large z are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
    10: Bibliography R
  • J. Raynal (1979) On the definition and properties of generalized 6 - j  symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
  • M. Razaz and J. L. Schonfelder (1980) High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.
  • M. D. Rogers (2005) Partial fractions expansions and identities for products of Bessel functions. J. Math. Phys. 46 (4), pp. 043509–1–043509–18.
  • D. H. Rouvray (1995) Combinatorics in Chemistry. In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.), pp. 1955–1981.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.