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11: 4.6 Power Series
§4.6 Power Series
§4.6(i) Logarithms
12: 14.32 Methods of Computation
In particular, for small or moderate values of the parameters μ and ν 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 Power Series
§10.65(iii) Cross-Products and Sums of Squares
§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 , are: …
Table 28.6.1: Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation.
n ρ n ( 1 ) ρ n ( 2 ) ρ n ( 3 )
§28.6(ii) Functions ce n and se n
Leading terms of the power series for the normalized functions are: …
15: 7.6 Series Expansions
§7.6(i) Power Series
16: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
17: 33.23 Methods of Computation
The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii ρ 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 - η , μ ( 2 ρ ) for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …
18: 33.19 Power-Series Expansions in r
§33.19 Power-Series Expansions in r
19: 33.6 Power-Series Expansions in ρ
§33.6 Power-Series Expansions in ρ
20: 3.10 Continued Fractions
§3.10(ii) Relations to Power Series
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 2 n - 1 , n = 1 , 2 , 3 , . …