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1: 13.4 Integral Representations
13.4.2 𝐌 ( a , b , z ) = 1 Γ ( b c ) 0 1 𝐌 ( a , c , z t ) t c 1 ( 1 t ) b c 1 d t , b > c > 0 ,
13.4.6 U ( a , b , z ) = ( 1 ) n z 1 b n Γ ( 1 + a b ) 0 𝐌 ( b a , b , t ) e t t b + n 1 t + z d t , | ph z | < π , n = 0 , 1 , 2 , , b < n < 1 + ( a b ) ,
2: 10.43 Integrals
§10.43(ii) Integrals over the Intervals ( 0 , x ) and ( x , )
3: 13.16 Integral Representations
13.16.2 M κ , μ ( z ) = Γ ( 1 + 2 μ ) z λ Γ ( 1 + 2 μ 2 λ ) Γ ( 2 λ ) 0 1 M κ λ , μ λ ( z t ) e 1 2 z ( t 1 ) t μ λ 1 2 ( 1 t ) 2 λ 1 d t , μ + 1 2 > λ > 0 ,
4: 9.12 Scorer Functions
If ζ = 2 3 z 3 / 2 or 2 3 x 3 / 2 , and K 1 / 3 is the modified Bessel function (§10.25(ii)), then …
5: 36.9 Integral Identities
For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). …
6: 10.22 Integrals
§10.22(iii) Integrals over the Interval ( x , )
Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …
7: 2.8 Differential Equations with a Parameter
and for simplicity ξ is assumed to range over a finite or infinite interval ( α 1 , α 2 ) with α 1 < 0 , α 2 > 0 . …
8: 18.39 Applications in the Physical Sciences
where x is a spatial coordinate, m the mass of the particle with potential energy V ( x ) , = h / ( 2 π ) is the reduced Planck’s constant, and ( a , b ) a finite or infinite interval. …
9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Two elements u and v in V are orthogonal if u , v = 0 . … where the infinite sum means convergence in norm, … Let X = [ a , b ] or [ a , b ) or ( a , b ] or ( a , b ) be a (possibly infinite, or semi-infinite) interval in . … Let X = ( a , b ) be a finite or infinite open interval in . … We assume a continuous spectrum λ 𝝈 c = [ 0 , ) , and a finite or countably infinite point spectrum 𝝈 p with elements λ n . …
10: 2.3 Integrals of a Real Variable
If, in addition, q ( t ) is infinitely differentiable on [ 0 , ) and … assume a and b are finite, and q ( t ) is infinitely differentiable on [ a , b ] . …Alternatively, assume b = , q ( t ) is infinitely differentiable on [ a , ) , and each of the integrals e i x t q ( s ) ( t ) d t , s = 0 , 1 , 2 , , converges as t uniformly for all sufficiently large x . … Assume that q ( t ) again has the expansion (2.3.7) and this expansion is infinitely differentiable, q ( t ) is infinitely differentiable on ( 0 , ) , and each of the integrals e i x t q ( s ) ( t ) d t , s = 0 , 1 , 2 , , converges at t = , uniformly for all sufficiently large x . …
  • (a)

    On ( a , b ) , p ( t ) and q ( t ) are infinitely differentiable and p ( t ) > 0 .