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power-series expansions in r

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1: 33.19 Power-Series Expansions in r
§33.19 Power-Series Expansions in r
2: 33.20 Expansions for Small | ϵ |
§33.20(ii) Power-Series in ϵ for the Regular Solution
3: 27.13 Functions
Mordell (1917) notes that r k ( n ) is the coefficient of x n in the power-series expansion of the k th power of the series for ϑ ( x ) . …
4: 33.23 Methods of Computation
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii ρ and r , respectively, and may be used to compute the regular and irregular solutions. Cancellation errors increase with increases in ρ and | r | , and may be estimated by comparing the final sum of the series with the largest partial sum. … Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions3.11(iv)) for the analytic continuations of Coulomb functions. Noble (2004) obtains double-precision accuracy for W η , μ ( 2 ρ ) for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …
5: 19.19 Taylor and Related Series
§19.19 Taylor and Related Series
The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … If n = 2 , then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … Then T N has at most one term if N 5 in the series for R F . …
6: 19.36 Methods of Computation
If (19.36.1) is used instead of its first five terms, then the factor ( 3 r ) 1 / 6 in Carlson (1995, (2.2)) is changed to ( 3 r ) 1 / 8 . … The incomplete integrals R F ( x , y , z ) and R G ( x , y , z ) can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to R C , accompanied by two quadratically convergent series in the case of R G ; compare Carlson (1965, §§5,6). … If the iteration of (19.36.6) and (19.36.12) is stopped when c s < r t s ( M and T being approximated by a s and t s , and the infinite series being truncated), then the relative error in R F and R G is less than r if we neglect terms of order r 2 . … For series expansions of Legendre’s integrals see §19.5. Faster convergence of power series for K ( k ) and E ( k ) can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12). …
7: Bibliography L
  • T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.
  • Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
  • J. L. López and E. Pérez Sinusía (2014) New series expansions for the confluent hypergeometric function M ( a , b , z ) . Appl. Math. Comput. 235, pp. 26–31.
  • J. L. López and N. M. Temme (2013) New series expansions of the Gauss hypergeometric function. Adv. Comput. Math. 39 (2), pp. 349–365.
  • Y. L. Luke (1959) Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.
  • 8: 1.10 Functions of a Complex Variable
    The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. … This singularity is removable if a n = 0 for all n < 0 , and in this case the Laurent series becomes the Taylor series. … (In other words n F n is the coefficient of ( z z 0 ) 1 in the Laurent expansion of 1 / ( f ( z ) f ( z 0 ) ) n in powers of ( z z 0 ) ; compare §1.10(iii).) … Let F ( x , z ) have a converging power series expansion of the form …
    9: 19.5 Maclaurin and Related Expansions
    §19.5 Maclaurin and Related Expansions
    Series expansions of F ( ϕ , k ) and E ( ϕ , k ) are surveyed and improved in Van de Vel (1969), and the case of F ( ϕ , k ) is summarized in Gautschi (1975, §1.3.2). For series expansions of Π ( ϕ , α 2 , k ) when | α 2 | < 1 see Erdélyi et al. (1953b, §13.6(9)). …
    10: 3.11 Approximation Techniques
    §3.11(ii) Chebyshev-Series Expansions
    In fact, (3.11.11) is the Fourier-series expansion of f ( cos θ ) ; compare (3.11.6) and §1.8(i). … For further details on Chebyshev-series expansions in the complex plane, see Mason and Handscomb (2003, §5.10). … be a formal power series. … When F has an explicit power-series expansion a possible choice of R is a Padé approximation to F . …