Let Pmk(y) be the Legendre
function of the first kind and let Γ(z) be the Gamma function. Then the general
Mehler-Fock transform of complex order k of a function g(y) is defined by the
equation
the
inversion theorem states
It is stated on page 416 of I. N. Sneddon’s book ‘The Use of Integral Transforms,
(1972) that apparently a class of functions g(y) for which this result is valid is not yet
clearly defined. The purpose of this paper is to define a class of functions g(y) as well
as a class f(x) and give proofs that the above inversion formula hold for these
classes.
