Abstract

An integral transform involving the associated Legendre function of zero order, P12+iτ(x), x[1,), as the kernel (considered as a function of τ), is called Mehler-Fock transform. Some generalizations, involving the function P12+iτμ(x), where the order μ is an arbitrary complex number, including the case when μ=0,1,2, have been known for some time. In this present note, we define a general Mehler-Fock transform involving, as the kernel, the Legendre function P12+tμ(x), of general order μ and an arbitrary index 12+t, t=σ+iτ, <τ<. Then we develop a symmetric inversion formulae for these transforms. Many well-known results are derived as special cases of this general form. These transforms are widely used for solving many axisymmetric potential problems.