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1: 22.16 Related Functions
Approximations for Small k , k
2: 11.1 Special Notation
x

real variable.

k

nonnegative integer.

δ

arbitrary small positive constant.

For the functions J ν ( z ) , Y ν ( z ) , H ν ( 1 ) ( z ) , H ν ( 2 ) ( z ) , I ν ( z ) , and K ν ( z ) see §§10.2(ii), 10.25(ii). The functions treated in this chapter are the Struve functions 𝐇 ν ( z ) and 𝐊 ν ( z ) , the modified Struve functions 𝐋 ν ( z ) and 𝐌 ν ( z ) , the Lommel functions s μ , ν ( z ) and S μ , ν ( z ) , the Anger function 𝐉 ν ( z ) , the Weber function 𝐄 ν ( z ) , and the associated Anger–Weber function 𝐀 ν ( z ) .
3: 11.6 Asymptotic Expansions
11.6.1 𝐊 ν ( z ) 1 π k = 0 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | π δ ,
11.6.2 𝐌 ν ( z ) 1 π k = 0 ( 1 ) k + 1 Γ ( k + 1 2 ) ( 1 2 z ) ν 2 k 1 Γ ( ν + 1 2 k ) , | ph z | 1 2 π δ .
11.6.3 0 z 𝐊 0 ( t ) d t 2 π ( ln ( 2 z ) + γ ) 2 π k = 1 ( 1 ) k + 1 ( 2 k ) ! ( 2 k 1 ) ! ( k ! ) 2 ( 2 z ) 2 k , | ph z | π δ ,
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 .
4: 8.20 Asymptotic Expansions of E p ( z )
5: 8.1 Special Notation
x

real variable.

k , n

nonnegative integers.

δ

arbitrary small positive constant.

6: 30.1 Special Notation
x

real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, 1 < x < 1 .

k

integer.

δ

arbitrary small positive constant.

7: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in 𝐱 space, across which Ψ K ( 𝐱 ; k ) or Ψ ( U ) ( 𝐱 ; k ) acquires an exponentially-small asymptotic contribution (in k ), associated with a complex critical point of Φ K or Φ ( U ) . …
8: 8.12 Uniform Asymptotic Expansions for Large Parameter
8.12.15 Q ( a , a ) 1 2 + 1 2 π a k = 0 c k ( 0 ) a k , | ph a | π δ ,
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: 10.17 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 ,
10.17.9 J ν ( z ) ( 2 π z ) 1 2 ( sin ω k = 0 ( 1 ) k b 2 k ( ν ) z 2 k + cos ω k = 0 ( 1 ) k b 2 k + 1 ( ν ) z 2 k + 1 ) , | ph z | π δ ,
10: 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 .