There is a one-to-one correspondence between the renewal function and the probability density function of the observation times of the corresponding events of each renewal process. In addition, the practical application of renewal processes requires information on the renewal function, and this function plays a fundamental role in studying the behavior of renewal processes, so that the necessary predictions can be made. In this paper, we determine the renewal function corresponding to two important distributions when the observation times have an Erlang distribution (n, lambda) and also when these times have a uniform distribution in the interval (0, b) using the Laplace transform and its related theorems. Finally, an application of the renewal process in physics will be presented.