power-series expansions in r
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1: 33.19 Power-Series Expansions in
§33.19 Power-Series Expansions in
…2: 33.20 Expansions for Small
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§33.20(ii) Power-Series in for the Regular Solution
…3: 27.13 Functions
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►Mordell (1917) notes that is the coefficient of
in the power-series expansion of the th power of the series for .
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4: 33.23 Methods of Computation
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►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii and , respectively, and may be used to compute the regular and irregular solutions.
Cancellation errors increase with increases in
and , and may be estimated by comparing the final sum of the series with the largest partial sum.
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►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.
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►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.
►Noble (2004) obtains double-precision accuracy for for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7).
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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 , then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … ►Then has at most one term if in the series for . … ►6: 19.36 Methods of Computation
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►If (19.36.1) is used instead of its first five terms, then the factor
in Carlson (1995, (2.2)) is changed to .
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►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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►If the iteration of (19.36.6) and (19.36.12) is stopped when ( and being approximated by and , and the infinite series being truncated), then the relative error in
and is less than if we neglect terms of order .
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►For series expansions of Legendre’s integrals see §19.5.
Faster convergence of power series for and can be achieved by using (19.5.1) and (19.5.2) in the right-hand sides of (19.8.12).
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7: Bibliography L
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New method to obtain small parameter power series expansions of Mathieu radial and angular functions.
Math. Comp. 78 (265), pp. 255–274.
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Integral and series representations of the Dirac delta function.
Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
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New series expansions for the confluent hypergeometric function
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Appl. Math. Comput. 235, pp. 26–31.
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New series expansions of the Gauss hypergeometric function.
Adv. Comput. Math. 39 (2), pp. 349–365.
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Expansion of the confluent hypergeometric function in series of Bessel functions.
Math. Tables Aids Comput. 13 (68), pp. 261–271.
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8: 1.10 Functions of a Complex Variable
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►The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus.
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►This singularity is removable if for all , and in this case the Laurent series becomes the Taylor series.
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►(In other words is the coefficient of
in the Laurent expansion of
in powers of ; compare §1.10(iii).)
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►Let have a converging power series expansion of the form
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9: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
… ► … ►Series expansions of and are surveyed and improved in Van de Vel (1969), and the case of is summarized in Gautschi (1975, §1.3.2). For series expansions of when see Erdélyi et al. (1953b, §13.6(9)). …10: 3.11 Approximation Techniques
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