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Bessel-function expansion

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1: 10.23 Sums
Partial Fractions
§10.23(iii) Series Expansions of Arbitrary Functions
Neumann’s Expansion
Fourier–Bessel Expansion
2: 10.35 Generating Function and Associated Series
Jacobi–Anger expansions: for z , θ , …
3: 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. …
4: 10.74 Methods of Computation
In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … For applications of the continued-fraction expansions (10.10.1), (10.10.2), (10.33.1), and (10.33.2) to the computation of Bessel functions and modified Bessel functions see Gargantini and Henrici (1967), Amos (1974), Gautschi and Slavik (1978), Tretter and Walster (1980), Thompson and Barnett (1986), and Cuyt et al. (2008). …
Fourier–Bessel Expansion
5: 33.20 Expansions for Small | ϵ |
§33.20(i) Case ϵ = 0
where A ( ϵ , ) is given by (33.14.11), (33.14.12), and
33.20.8 H k ( ; r ) = p = 2 k 3 k ( 2 r ) ( p + 1 ) / 2 C k , p Y 2 + 1 + p ( 8 r ) , r > 0 ,
§33.20(iv) Uniform Asymptotic Expansions
These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders 2 + 1 and 2 + 2 .
6: 6.10 Other Series Expansions
§6.10(ii) Expansions in Series of Spherical Bessel Functions
7: 10.41 Asymptotic Expansions for Large Order
§10.41 Asymptotic Expansions for Large Order
§10.41(i) Asymptotic Forms
§10.41(ii) Uniform Expansions for Real Variable
8: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
§10.19(i) Asymptotic Forms
§10.19(ii) Debye’s Expansions
For error bounds for the first of (10.19.6) see Olver (1997b, p. 382).
§10.19(iii) Transition Region
9: 13.24 Series
§13.24(ii) Expansions in Series of Bessel Functions
Additional expansions in terms of Bessel functions are given in Luke (1959). …
10: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
Products
§10.40(ii) Error Bounds for Real Argument and Order
§10.40(iii) Error Bounds for Complex Argument and Order
§10.40(iv) Exponentially-Improved Expansions