A procedure for transforming a series whose lth term is the product of the lth Legendre polynomial and a coefficient, a_l , having certain properties, into the sum of a contour integral of a(l ) and terms involving poles of a(l ), where a(l ) is a meromorphic function such that a(l ) equals a_l at integral values of l; used in studying rainbows, propagation of radio waves around the earth, scattering from various potentials, and scattering of elementary particles. Also known as Sommerfeld-Watson transformation.