Fourier Coefficients of Mathieu Functions Stable Regions *

Continued fraction expansions for the eigenvalues A= A(I', h2 ) and for the Fourier coefficients c~n/co are given by Tamir and Wang in [2]. These relations are valid both in the s table (I' real) and unstable (I' complex) regions . F (I') == C~,.1Co is numerically evaluated and depicted. It is seen from the graphs in [2] that the given formulas for C~,JCO are not particularly suitable for numerical evaluation in the stable regions. Representations for F(I') = c~,Jco derived from other numerical methods will now be introdu ced. The eigenvalues A of Mathieu's differential equation can be determined as the eigenvalues of the symmetrical tridiagonal matrix of the homogeneous system of eq (3) with the help of the bisection method [3]. The n the Fourier coeffi cients are obtainable from the recurrence formula (3) under consideration of stability (Miller's recurrence algorithm [4,5]). The accuracy of this method depends principally upon the number of equations used. Numerical work was carried out to single precision on the computer of the Technische Hochschule Darmstadt. For evaluating the eigenvalues A, a modified version of the bisection method [6] was used. This modification prohibits overflow. Also, a numerical test ensures that the trunca-