About the Project

trigonometric%20series%20expansions

AdvancedHelp

(0.004 seconds)

10 matching pages

1: 6.20 Approximations
  • 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
  • Luke and Wimp (1963) covers Ei ( x ) for x 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function (§13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.

  • 2: 25.12 Polylogarithms
    The cosine series in (25.12.7) has the elementary sum … For real or complex s and z the polylogarithm Li s ( z ) is defined by … For each fixed complex s the series defines an analytic function of z for | z | < 1 . The series also converges when | z | = 1 , provided that s > 1 . … The notation ϕ ( z , s ) was used for Li s ( z ) in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. …
    3: 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
  • Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for ( 1 + 2 x ) e x 2 erfc x on ( 0 , ) (22D).

  • §7.24(iii) Padé-Type Expansions
    4: 6.16 Mathematical Applications
    Consider the Fourier seriesThe first maximum of 1 2 Si ( x ) for positive x occurs at x = π and equals ( 1.1789 ) × 1 4 π ; compare Figure 6.3.2. …Compare Figure 6.16.1. … It occurs with Fourier-series expansions of all piecewise continuous functions. … …
    5: Bibliography F
  • S. Fempl (1960) Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
  • W. B. Ford (1960) Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.
  • C. L. Frenzen and R. Wong (1986) Asymptotic expansions of the Lebesgue constants for Jacobi series. Pacific J. Math. 122 (2), pp. 391–415.
  • 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.
  • T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.
  • 6: 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.
  • M. Katsurada (2003) Asymptotic expansions of certain q -series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.
  • 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.
  • 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.
  • 7: 5.11 Asymptotic Expansions
    §5.11 Asymptotic Expansions
    Wrench (1968) gives exact values of g k up to g 20 . … The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). …
    8: Bibliography M
  • H. Maass (1971) Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.
  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
  • L. J. Mordell (1958) On the evaluation of some multiple series. J. London Math. Soc. (2) 33, pp. 368–371.
  • C. Mortici (2011a) A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.
  • 9: Bibliography S
  • M. J. Seaton (2002b) FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.
  • H. Shanker (1939) On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.
  • H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
  • B. D. Sleeman (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.
  • F. Stenger (1993) Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
  • 10: 25.11 Hurwitz Zeta Function
    The function ζ ( s , a ) was introduced in Hurwitz (1882) and defined by the series expansion
    §25.11(iv) Series Representations
    For other series expansions similar to (25.11.10) see Coffey (2008). …
    §25.11(x) Further Series Representations
    §25.11(xii) a -Asymptotic Behavior