series representation
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21: 14.32 Methods of Computation
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►In particular, for small or moderate values of the parameters and 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.
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22: Bibliography B
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Riemann zeta function: Rapidly converging series and integral representations.
Appl. Math. Lett. 5 (2), pp. 83–88.
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23: 34.6 Definition: Symbol
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►The symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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24: Bibliography S
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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.
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25: 19.36 Methods of Computation
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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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►Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals.
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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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26: Bibliography M
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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation.
SIAM J. Numer. Anal. 3 (3), pp. 390–409.
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27: 16.15 Integral Representations and Integrals
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►These representations can be used to derive analytic continuations of the Appell functions, including convergent series expansions for large , large , or both.
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28: 22.15 Inverse Functions
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§22.15(ii) Representations as Elliptic Integrals
… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). For power-series expansions see Carlson (2008).29: 21.10 Methods of Computation
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►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemann surface.
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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.