The Jost function method is extended to study scattering phenomena produced by potentials having spheroidal symmetry. The spheroidal radial functions are constructed by taking the spherical wave functions as bases.

The role of the Jost function in spherical potential scattering is reviewed. The relation between zeros of the Jost function and the formation of bound and resonant states is then established for spheroidal potentials. The domain of analyticity of the spheroidal Jost function is studied for three classes of basis functions: those belonging to spherical potentials having only a bounded first and second moments, those belonging to a Yukawa potential, and those belonging to truncated potentials.