Hankel complementary integral transformations of arbitrary order

Abstract

Four selfreciprocal integral transformations of Hankel type are defined through [formula omitted] where i = 1, 2, 3, 4; μ ≥ 0; α1(x) = x1 +2μ, g1,μ(x) = x−μJμ(x), Jμ(x) being the Bessel function of the first kind of order; μ; α2(x) = x1−2μ, g2,μ(x) =(−1)μx2μ g1,μ(x); α3(x) = x−1−2μ, g3,μ(x) = x1+2μ g1,μ(x), and α4(x) = x−1+2μ, g4,μ(x) = (−1)μx g1,μ(x). The simultaneous use of transformations H1,μ and H2,μ (which are denoted by Hμ) allows us to solve many problems of Mathematical Physics involving the differential operator Δμ = D2 + (1 + 2μ)x−1D, whereas the pair of transformations H3,μ and H4,μ (which we express by Hμ) permits us to tackle those problems containing its adjoint operator [formula omitted], no matter what the real value of μ be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation [formula omitted], which is now valid for all real μ. © 1987, Hindawi Publishing Corporation. All rights reserved.