On the Convergence and Approximation of Integrated Semigroups

We extend the well known theorem on convergence and approximation of C0-semigroups due to H. F. Trotter and T. Kato to the context of integrated semigroups. The result is obtained as a consequence of an "integrated version" of the Trotter-Kato theorem based on the Laplace transform. An application to the complete second order equation is given.