Fractional Integration and Dual Integral Equations

In the analysis of mixed boundary value problems by the use of Hankel transforms we often encounter pairs of dual integral equations which can be written in the symmetrical form (1.1) Equations of this type seem to have been formulated first by Weber in his paper (1) in which he derives (by inspection) the solution for the case in which α — β = ½, v = 0, F ≡ 1, G ≡ 0. The first direct solution of a pair of equations of this type was given by Beltrami (2) for the same values of α — β and v with G(p) ≡ 0 but with F(ρ) arbitrary.