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in terms of Bessel functions of fixed order

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1: 2.8 Differential Equations with a Parameter
§2.8(iv) Case III: Simple Pole
For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …
2: 33.12 Asymptotic Expansions for Large η
The second set is in terms of Bessel functions of orders 2 + 1 and 2 + 2 , and they are uniform for fixed and 0 z 1 - δ , where δ again denotes an arbitrary small positive constant. …
3: 10.68 Modulus and Phase Functions
where M ν ( x ) ( > 0 ) , N ν ( x ) ( > 0 ) , θ ν ( x ) , and ϕ ν ( x ) are continuous real functions of x and ν , with the branches of θ ν ( x ) and ϕ ν ( x ) chosen to satisfy (10.68.18) and (10.68.21) as x . … Equations (10.68.8)–(10.68.14) also hold with the symbols ber , bei , M , and θ replaced throughout by ker , kei , N , and ϕ , respectively. In place of (10.68.7), … When ν is fixed, μ = 4 ν 2 , and x Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …
4: 18.15 Asymptotic Approximations
The latter expansions are in terms of Bessel functions, and are uniform in complex z -domains not containing neighborhoods of 1. … These expansions are in terms of Bessel functions and modified Bessel functions, respectively. …
In Terms of Elementary Functions
In Terms of Bessel Functions
In Terms of Airy Functions
5: 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
§10.19(iii) Transition Region
See also §10.20(i).
6: 10.40 Asymptotic Expansions for Large Argument
ν -Derivative
§10.40(ii) Error Bounds for Real Argument and Order
For the error term in (10.40.1) see §10.40(iii).
§10.40(iii) Error Bounds for Complex Argument and Order
For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).
7: 14.15 Uniform Asymptotic Approximations
§14.15(i) Large μ , Fixed ν
Here I and K are the modified Bessel functions10.25(ii)). … For asymptotic expansions and explicit error bounds, see Dunster (2003b).
§14.15(iii) Large ν , Fixed μ
See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials P n ( cos θ ) as n with θ fixed. …
8: 13.9 Zeros
When a < 0 and b > 0 let ϕ r , r = 1 , 2 , 3 , , be the positive zeros of M ( a , b , x ) arranged in increasing order of magnitude, and let j b - 1 , r be the r th positive zero of the Bessel function J b - 1 ( x ) 10.21(i)). … For fixed a , b the large z -zeros of M ( a , b , z ) satisfy … For fixed b and z in the large a -zeros of M ( a , b , z ) are given by … For fixed a and z in the function M ( a , b , z ) has only a finite number of b -zeros. … For fixed b and z in the large a -zeros of U ( a , b , z ) are given by …
9: 10.17 Asymptotic Expansions for Large Argument
§10.17 Asymptotic Expansions for Large Argument
§10.17(iii) Error Bounds for Real Argument and Order
If these expansions are terminated when k = - 1 , then the remainder term is bounded in absolute value by the first neglected term, provided that max ( ν - 1 2 , 1 ) . … For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).
10: 33.20 Expansions for Small | ϵ |
§33.20(i) Case ϵ = 0
The functions J and I are as in §§10.2(ii), 10.25(ii), and the coefficients C k , p are given by C 0 , 0 = 1 , C 1 , 0 = 0 , and … The functions Y and K are as in §§10.2(ii), 10.25(ii), and the coefficients C k , p are given by (33.20.6).
§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 .