International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 171-180

The Mehler-Fock transform of general order and arbitrary index and its inversion

Cyril Nasim

Department of Mathematics and Statistics, The University of Calgary, Alberta, Calgary, Canada

Received 24 July 1983

Copyright © 1984 Cyril Nasim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.