series expansions

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21: 9.19 Approximations

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§9.19(ii) Expansions in Chebyshev Series

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Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ .

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MacLeod (1994) supplies Chebyshev-series expansions to cover $\operatorname{Gi}\left(x\right)$ for $0\leq x<\infty$ and $\operatorname{Hi}\left(x\right)$ for $-\infty<x\leq 0$ . The Chebyshev coefficients are given to 20D.

22: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii

\rho

and

r

, respectively, and may be used to compute the regular and irregular solutions. … ►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 expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. ►Noble (2004) obtains double-precision accuracy for

W_{-\eta,\mu}\left(2\rho\right)

for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

23: 13.11 Series

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(n+1)A_{n+1}=(n+b-1)A_{n-1}+(2a-b)A_{n-2},

n=2,3,4,\dots

… ►For other series expansions see Tricomi (1954, §1.8), Hansen (1975, §§66 and 87), Prudnikov et al. (1990, §6.6), López and Temme (2010a) and López and Pérez Sinusía (2014). …

24: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

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25: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

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26: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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27: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ► … ►Series expansions of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

are surveyed and improved in Van de Vel (1969), and the case of

F\left(\phi,k\right)

is summarized in Gautschi (1975, §1.3.2). For series expansions of

\Pi\left(\phi,\alpha^{2},k\right)

when

|\alpha^{2}|<1

see Erdélyi et al. (1953b, §13.6(9)). …

28: 28.15 Expansions for Small $q$

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§28.15(i) Eigenvalues $\lambda_{\nu}\left(q\right)$

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29: 28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions

§28.19 Expansions in Series of $\operatorname{me}_{\nu+2n}$ Functions

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30: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …