# power series

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## 11—20 of 96 matching pages

##### 12: 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. …
##### 13: 10.65 Power Series
###### §10.65(iv) Compendia
For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).
##### 14: 28.6 Expansions for Small $q$
###### §28.6(i) Eigenvalues
Leading terms of the of the power series for $m=7,8,9,\dots$ are: …
###### §28.6(ii) Functions $\mathrm{ce}_{n}$ and $\mathrm{se}_{n}$
Leading terms of the power series for the normalized functions are: …
##### 17: 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). …
##### 20: 3.10 Continued Fractions
###### Stieltjes Fractions
We say that it corresponds to the formal power seriesFor several special functions the $S$-fractions are known explicitly, but in any case the coefficients $a_{n}$ can always be calculated from the power-series coefficients by means of the quotient-difference algorithm; see Table 3.10.1. … We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_{n}$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\dots$. …