Hankel Complementary Integral Transformations of Arbitrary Order M . Linares Linares

Four selfreciprocal integral transformations of Hankel type are defined through I,Cy) fo’,Cx),,,Cxy)Cx, ’ where 1,2,3, 4; Ix 0; ct(x) x /, g,(x) x-J(x), J(x) being the Bessel function of the first kind oforder Ix; th(x) x1-2, g2,(x) (-1)x2gt,,(x); tx3(x) x--2, ,qx,(x) x /,q,,(x), and ct,(x) x 2, g4,(x)(-1)xgLa(x). The simultaneous use of transformations Y/’I, and /’z, (which are denoted by allows us to solve many problems of Mathematical Physics involving the differential operator A D + (1 + 2Ix)x-|D, whereas the pair of transformations /’,, and /’4, (which we express by) permits us to tackle those problems containing its adjoint operator AD2-(1 + 2Ix)x-tD + (1 + 2it)x-2, no matter what the real value of It be. These transformations are also investigated in a space of generalized functions according to the mixed Parseval equation fo’Y x)g x fo’ aO which is now valid for all real It.