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11: 19.19 Taylor and Related Series
§19.19 Taylor and Related Series
12: 7.17 Inverse Error Functions
§7.17(ii) Power Series
13: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
14: 8.7 Series Expansions
§8.7 Series Expansions
15: 28.6 Expansions for Small q
§28.6(i) Eigenvalues
28.6.19 a ( 2 n + 2 ) 2 q 2 a ( 2 n ) 2 q 2 a ( 2 n 2 ) 2 q 2 a 2 2 = q 2 ( 2 n + 4 ) 2 a q 2 ( 2 n + 6 ) 2 a , a = b 2 n + 2 ( q ) .
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 )
16: 11.13 Methods of Computation
Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of z , they are cumbersome to use when | z | is large owing to slowness of convergence and cancellation. …
17: 30.4 Functions of the First Kind
§30.4(iii) Power-Series Expansion
18: 11.2 Definitions
§11.2(i) Power-Series Expansions
19: 12.15 Generalized Parabolic Cylinder Functions
See Faierman (1992) for power series and asymptotic expansions of a solution of (12.15.1).
20: 10.74 Methods of Computation
The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument x or z is sufficiently small in absolute value. … In the interval 0 < x < ν , J ν ( x ) needs to be integrated in the forward direction and Y ν ( x ) in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). …