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asymptotic expansions as ϵ→0

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1: 2.2 Transcendental Equations
2.2.7 f ( x ) x + f 0 + f 1 x 1 + f 2 x 2 + , x .
2.2.8 x y F 0 F 1 y 1 F 2 y 2 , y ,
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …
2: 2.1 Definitions and Elementary Properties
2.1.13 f ( x ) = s = 0 n 1 a s x s + O ( x n )
2.1.14 f ( x ) a 0 + a 1 x 1 + a 2 x 2 + , x in 𝐗 .
2.1.16 f ( x ) a 0 + a 1 ( x c ) + a 2 ( x c ) 2 + , x c in 𝐗 ,
As an example, in the sector | ph z | 1 2 π δ ( < 1 2 π ) each of the functions 0 , e z , and e z (principal value) has the null asymptotic expansion
2.1.18 f ( u , x ) s = 0 a s ( u ) x s
3: 33.20 Expansions for Small | ϵ |
§33.20(iii) Asymptotic Expansion for the Irregular Solution
§33.20(iv) Uniform Asymptotic Expansions
For a comprehensive collection of asymptotic expansions that cover f ( ϵ , ; r ) and h ( ϵ , ; r ) as ϵ 0 ± and are uniform in r , including unbounded values, see Curtis (1964a, §7). …
4: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). … If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . These asymptotic expansions are uniform with respect to z , including cut neighborhoods of z 0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
5: 33.11 Asymptotic Expansions for Large ρ
33.11.1 H ± ( η , ρ ) e ± i θ ( η , ρ ) k = 0 ( a ) k ( b ) k k ! ( ± 2 i ρ ) k ,
f ( η , ρ ) k = 0 f k ,
g ( η , ρ ) k = 0 g k ,
f ^ ( η , ρ ) k = 0 f ^ k ,
g ^ ( η , ρ ) k = 0 g ^ k ,
6: 4.13 Lambert W -Function
4.13.11 W ± 1 ( x 0 i ) η ln η + n = 1 1 η n m = 1 n [ n n m + 1 ] ( ln η ) m m ! ,
7: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For asymptotic expansions of ϕ ( ρ , β ; z ) as z in various sectors of the complex z -plane for fixed real values of ρ and fixed real or complex values of β , see Wright (1935) when ρ > 0 , and Wright (1940b) when 1 < ρ < 0 . For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when ρ > 0 , and Wong and Zhao (1999a) when 1 < ρ < 0 . …
8: 7.17 Inverse Error Functions
§7.17(ii) Power Series
where a 0 = 1 and the other coefficients follow from the recursion …
§7.17(iii) Asymptotic Expansion of inverfc x for Small x
As x 0
7.17.3 inverfc x u 1 / 2 + a 2 u 3 / 2 + a 3 u 5 / 2 + a 4 u 7 / 2 + ,
9: 2.3 Integrals of a Real Variable
2.3.2 0 e x t q ( t ) d t s = 0 q ( s ) ( 0 ) x s + 1 , x + .
2.3.7 q ( t ) s = 0 a s t ( s + λ μ ) / μ , t 0 + ,
  • (b)

    As t a +

    2.3.14
    p ( t ) p ( a ) + s = 0 p s ( t a ) s + μ ,
    q ( t ) s = 0 q s ( t a ) s + λ 1 ,

    and the expansion for p ( t ) is differentiable. Again λ and μ are positive constants. Also p 0 > 0 (consistent with (a)).

  • 2.3.16 q ( t ) p ( t ) s = 0 b s v ( s + λ μ ) / μ , v 0 + ,
  • (b)

    As t a + the asymptotic expansions (2.3.14) apply, and each is infinitely differentiable. Again λ , μ , and p 0 are positive.

  • 10: 6.12 Asymptotic Expansions
    For these and other error bounds see Olver (1997b, pp. 109–112) with α = 0 . …