Acceleration of the convergence of series containing Mathieu functions using Shanks transformation

A modification of the standard application of Shanks transformation is shown to improve the convergence rate in certain cases where the straightforward application of Shanks transformation fails. Here, the straightforward application of Shanks transformation to a well known series expansion containing Mathieu functions failed to improve the convergence rate. However, convergence was achieved by a new method of applying Shanks transformation. This new method requires analysis of the behavior of the series terms to determine the cause of the slow or failing convergence. Then, the Shanks transformation was applied only to the slowly convergent part of the series. This work is important because, with this new method, convergence may be achieved in cases where the standard application of Shanks transformation fails to improve the convergence rate. The paper takes as a case study the electromagnetic problem of the expansion of a cylindrical wave in a series of Mathieu functions.