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asymptotic expansions for small parameters

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1: 2.5 Mellin Transform Methods
§2.5(iii) Laplace Transforms with Small Parameters
For examples in which the integral defining the Mellin transform h ( z ) does not exist for any value of z , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
2: 12.9 Asymptotic Expansions for Large Variable
12.9.1 U ( a , z ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , | ph z | 3 4 π - δ ( < 3 4 π ) ,
12.9.2 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s , | ph z | 1 4 π - δ ( < 1 4 π ) .
12.9.3 U ( a , z ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s ± i 2 π Γ ( 1 2 + a ) e i π a e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s , 1 4 π + δ ± ph z 5 4 π - δ ,
12.9.4 V ( a , z ) 2 π e 1 4 z 2 z a - 1 2 s = 0 ( 1 2 - a ) 2 s s ! ( 2 z 2 ) s ± i Γ ( 1 2 - a ) e - 1 4 z 2 z - a - 1 2 s = 0 ( - 1 ) s ( 1 2 + a ) 2 s s ! ( 2 z 2 ) s , - 1 4 π + δ ± ph z 3 4 π - δ .
3: 8.20 Asymptotic Expansions of E p ( z )
4: 10.17 Asymptotic Expansions for Large Argument
10.17.3 J ν ( z ) ( 2 π z ) 1 2 ( cos ω k = 0 ( - 1 ) k a 2 k ( ν ) z 2 k - sin ω k = 0 ( - 1 ) k a 2 k + 1 ( ν ) z 2 k + 1 ) , | ph z | π - δ ,
10.17.4 Y ν ( z ) ( 2 π z ) 1 2 ( sin ω k = 0 ( - 1 ) k a 2 k ( ν ) z 2 k + cos ω k = 0 ( - 1 ) k a 2 k + 1 ( ν ) z 2 k + 1 ) , | ph z | π - δ ,
10.17.11 H ν ( 1 ) ( z ) i ( 2 π z ) 1 2 e i ω k = 0 i k b k ( ν ) z k , - π + δ ph z 2 π - δ ,
5: 10.40 Asymptotic Expansions for Large Argument
10.40.3 I ν ( z ) e z ( 2 π z ) 1 2 k = 0 ( - 1 ) k b k ( ν ) z k , | ph z | 1 2 π - δ ,
10.40.5 I ν ( z ) e z ( 2 π z ) 1 2 k = 0 ( - 1 ) k a k ( ν ) z k ± i e ± ν π i e - z ( 2 π z ) 1 2 k = 0 a k ( ν ) z k , - 1 2 π + δ ± ph z 3 2 π - δ .
6: 8.11 Asymptotic Approximations and Expansions
8.11.6 γ ( a , z ) - z a e - z k = 0 ( - a ) k b k ( λ ) ( z - a ) 2 k + 1 , 0 < λ < 1 , | ph a | π 2 - δ .
8.11.7 Γ ( a , z ) z a e - z k = 0 ( - a ) k b k ( λ ) ( z - a ) 2 k + 1 , λ > 1 , | ph a | 3 π 2 - δ .
7: 11.6 Asymptotic Expansions
11.6.6 K ν ( λ ν ) ( 1 2 λ ν ) ν - 1 π Γ ( ν + 1 2 ) k = 0 k ! c k ( λ ) ν k , | ph ν | 1 2 π - δ ,
11.6.7 M ν ( λ ν ) - ( 1 2 λ ν ) ν - 1 π Γ ( ν + 1 2 ) k = 0 k ! c k ( i λ ) ν k , | ph ν | 1 2 π - δ .
8: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.16 e ± π i a 2 i sin ( π a ) Q ( - a , a e ± π i ) ± 1 2 - i 2 π a k = 0 c k ( 0 ) ( - a ) - k , | ph a | π - δ ,
9: 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).
10: 11.11 Asymptotic Expansions of Anger–Weber Functions