Numerical Review of Mathieu Function Programs for Integer Orders and Real Parameters

The Mathieu function is a special function satisfying the Mathieu differential equation. Since its inception in 1868, numerous algorithms and programs have been published to calculate it, and so it is about time to review the performance of available software. First, the fundamentals of Mathieu functions are summarized such as definition, normalization, nomenclature, and methods of solution. Then, we review several programs for Mathieu functions of integer orders with real parameters and compare the results numerically by running individual software; in addition, Bessel function routines are also compared. Finally, a straightforward algorithm is recommended with codes written in MATLAB and GNU Octave