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1: 6.4 Analytic Continuation
6.4.4 Ci ( z e ± π i ) = ± π i + Ci ( z ) ,
2: 6.2 Definitions and Interrelations
Hyperbolic Analogs of the Sine and Cosine Integrals
3: 18.17 Integrals
For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). …
18.17.8 ( H n ( x ) ) 2 + 2 n ( n ! ) 2 e x 2 ( V ( - n - 1 2 , 2 1 2 x ) ) 2 = 2 n + 3 2 n ! e x 2 π 0 e - ( 2 n + 1 ) t + x 2 tanh t ( sinh 2 t ) 1 2 d t .
4: 10.20 Uniform Asymptotic Expansions for Large Order
Note: Another way of arranging the above formulas for the coefficients A k ( ζ ) , B k ( ζ ) , C k ( ζ ) , and D k ( ζ ) would be by analogy with (12.10.42) and (12.10.46). …
10.20.17 z = ± ( τ coth τ - τ 2 ) 1 2 ± i ( τ 2 - τ tanh τ ) 1 2 , 0 τ τ 0 ,
where τ 0 = 1.19968 is the positive root of the equation τ = coth τ . …
5: 19.25 Relations to Other Functions
19.25.16 Π ( ϕ , α 2 , k ) = - 1 3 ω 2 R J ( c - 1 , c - k 2 , c , c - ω 2 ) + ( c - 1 ) ( c - k 2 ) ( α 2 - 1 ) ( 1 - ω 2 ) R C ( c ( α 2 - 1 ) ( 1 - ω 2 ) , ( α 2 - c ) ( c - ω 2 ) ) , ω 2 = k 2 / α 2 .
For analogous integrals of the second kind, which are not invertible in terms of single-valued functions, see (19.29.20) and (19.29.21) and compare with Gradshteyn and Ryzhik (2000, §3.153,1–10 and §3.156,1–9). …