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11: 8.20 Asymptotic Expansions of E p ( z )
§8.20 Asymptotic Expansions of E p ( z )
§8.20(i) Large z
For an exponentially-improved asymptotic expansion of E p ( z ) see §2.11(iii).
§8.20(ii) Large p
8.20.6 E p ( λ p ) e λ p ( λ + 1 ) p k = 0 A k ( λ ) ( λ + 1 ) 2 k 1 p k ,
12: 6.13 Zeros
Values of c 1 and c 2 to 30D are given by MacLeod (1996b). …
6.13.2 c k , s k α + 1 α 16 3 1 α 3 + 1673 15 1 α 5 5 07746 105 1 α 7 + ,
13: 33.11 Asymptotic Expansions for Large ρ
§33.11 Asymptotic Expansions for Large ρ
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 ,
14: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
Products
ν -Derivative
§10.40(iv) Exponentially-Improved Expansions
15: 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).
16: 28.8 Asymptotic Expansions for Large q
§28.8 Asymptotic Expansions for Large q
§28.8(ii) Sips’ Expansions
§28.8(iii) Goldstein’s Expansions
σ m 1 + s 2 3 h + 4 s 2 + 3 2 7 h 2 + 19 s 3 + 59 s 2 11 h 3 + ,
τ m + 1 2 h 1 4 s 2 s 2 + 3 2 6 h 7 s 3 + 47 s 2 10 h 2 ,
17: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
§11.6(i) Large | z | , Fixed ν
§11.6(ii) Large | ν | , Fixed z
More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions2.1(v)). …
18: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
and …
§5.11(ii) Error Bounds and Exponential Improvement
§5.11(iii) Ratios
19: 12.11 Zeros
§12.11(ii) Asymptotic Expansions of Large Zeros
§12.11(iii) Asymptotic Expansions for Large Parameter
12.11.4 u a , s 2 1 2 μ ( p 0 ( α ) + p 1 ( α ) μ 4 + p 2 ( α ) μ 8 + ) ,
12.11.7 u a , s 2 1 2 μ ( q 0 ( β ) + q 1 ( β ) μ 4 + q 2 ( β ) μ 8 + ) ,
12.11.9 u a , 1 2 1 2 μ ( 1 1.85575 708 μ 4 / 3 0.34438 34 μ 8 / 3 0.16871 5 μ 4 0.11414 μ 16 / 3 0.0808 μ 20 / 3 ) ,
20: 10.17 Asymptotic Expansions for Large Argument
§10.17 Asymptotic Expansions for Large Argument
§10.17(ii) Asymptotic Expansions of Derivatives
§10.17(iii) Error Bounds for Real Argument and Order
§10.17(v) Exponentially-Improved Expansions
For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).