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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: 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 .

  • 9: 1.4 Calculus of One Variable
    If f ( n ) exists and is continuous on an interval I , then we write f C n ( I ) . …When n is unbounded, f is infinitely differentiable on I and we write f C ( I ) . …
    Infinite Integrals
    With a < b , the total variation of f ( x ) on a finite or infinite interval ( a , b ) is …where the supremum is over all sets of points x 0 < x 1 < < x n in the closure of ( a , b ) , that is, ( a , b ) with a , b added when they are finite. …
    10: Bibliography I
  • IEEE (2015) IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015. The Institute of Electrical and Electronics Engineers, Inc..
  • IEEE (2018) IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017. The Institute of Electrical and Electronics Engineers, Inc..
  • Y. Ikebe, Y. Kikuchi, I. Fujishiro, N. Asai, K. Takanashi, and M. Harada (1993) The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J 0 ( z ) - i J 1 ( z ) and of Bessel functions J m ( z ) of any real order m . Linear Algebra Appl. 194, pp. 35–70.
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.