power-series expansion

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1: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

2: 12.4 Power-Series Expansions

§12.4 Power-Series Expansions

…

3: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters

\mu

and

\nu

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. …

4: 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. …

5: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

…

6: 33.6 Power-Series Expansions in $\rho$

§33.6 Power-Series Expansions in $\rho$

…

7: 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. … ►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). …

8: 28.15 Expansions for Small $q$

… ►

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

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9: 7.6 Series Expansions

… ►

§7.6(i) Power Series

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10: 30.3 Eigenvalues

… ►

§30.3(iv) Power-Series Expansion

…