Thel-th partial-waveS-matrix element for cut-off potentials is Taylor-expanded in the energy and momentum variables. The expansion is based on a suitable finite-interval Green’s function, by which the derivatives are obtained in a recursion form involving only repeated integrations of the Green’s function itself. The Taylor series can be analytically continued by resorting to the Pade method; some results of a numerical experiment are quoted. In this paper, however, the emphasis is put on a rather indirect but rigorous procedure based on the Hadamard theory, which allows us to determine, in principle, all the poles of theS-matrix starting from its momentum Taylor series. Numerical results are displayed.