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1: 13.24 Series
For expansions of arbitrary functions in series of M κ , μ ( z ) functions see Schäfke (1961b). …
2: 10.23 Sums
§10.23(iii) Series Expansions of Arbitrary Functions
For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). …
3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
1.18.16 lim m a b | f ( x ) n = 0 m c n ϕ n ( x ) | 2 d x = 0 .
4: 14.18 Sums
For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). …
5: 18.18 Sums
§18.18(i) Series Expansions of Arbitrary Functions
6: 18.24 Hahn Class: Asymptotic Approximations
This expansion is in terms of the parabolic cylinder function and its derivative. … This expansion is in terms of confluent hypergeometric functions. … The first expansion holds uniformly for δ x 1 + δ , and the second for 1 δ x 1 + δ 1 , δ being an arbitrary small positive constant. Both expansions are in terms of parabolic cylinder functions. … Taken together, these expansions are uniformly valid for < x < and for a in unbounded intervals—each of which contains [ 0 , ( 1 δ ) n ] , where δ again denotes an arbitrary small positive constant. …
7: 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 .
8: 7.18 Repeated Integrals of the Complementary Error Function
§7.18 Repeated Integrals of the Complementary Error Function
where n = 1 , 2 , 3 , , and A , B are arbitrary constants. …
Parabolic Cylinder Functions
§7.18(vi) Asymptotic Expansion
9: 8.11 Asymptotic Approximations and Expansions
where δ denotes an arbitrary small positive constant. …
10: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.8 𝐀 ν ( λ ν ) 1 π k = 0 ( 2 k ) ! a k ( λ ) ν 2 k + 1 , ν , | ph ν | π δ ,
11.11.10 𝐀 ν ( λ ν ) 1 π k = 0 ( 2 k ) ! a k ( λ ) ν 2 k + 1 , ν , | ph ν | π δ .
11.11.11 𝐀 ν ( λ ν ) ( 2 π ν ) 1 / 2 e ν μ k = 0 ( 1 2 ) k b k ( λ ) ν k , ν , | ph ν | π 2 δ ,
11.11.18 𝐉 ν ( ν ) 2 1 / 3 3 2 / 3 Γ ( 2 3 ) ν 1 / 3 , ν , | ph ν | π δ ,
11.11.19 𝐄 ν ( ν ) 2 1 / 3 3 7 / 6 Γ ( 2 3 ) ν 1 / 3 , ν , | ph ν | π δ .