Numerical aspects of Mathieu eigenvalues

Simple and double eigenvalues of the Hill operator with a two-term potential

“…Examples in the previous subsection show that in Lemmas 9, 10 and Theorem 11 the assumptions that a, b be real, or L be self-adjoint, are important. But let us follow [3,35,19,4,5,37] and raise a general question about the structure of spectral Riemann surfaces related to these problems. Of course, it would be interesting to change both and t in complex plane, i.e., to consider ( , t) ∈ C 2 but for a while, let us talk about fixed positive t. Define, for each t > 0, four surfaces .17), and H ± are defined by (3.44).…”

The double points of Mathieu’s differential equation

“…The continued-fraction method formed the basis for the present calculations. A full discussion of the method is given in [1]. In addition, a comprehensive code now exists [3] for obtaining all solutions of Mathieu's equation, including the eigenvalues, for q > 0.…”

“…For the sake of conciseness, the derivation of the particular continued fraction forms will not be repeated here. The availability of [1] will be assumed and only the necessary modifications will be explained below. In essence, there is a complex-valued function, say Tia, q), such that, a necessary and sufficient condition for a(g) to be an eigenvalue is that Tia, q) = 0.…”

Surface motion of a semi-elliptical hill for incident plane SH waves

“…If ETA and M are beyond some values, or q is very large, Mathieu functions calculation subroutines (e.g., Blanch, 1966;Leeb, 1979;Shirts, 1993a, b;Alhargan, 2000aAlhargan, , b, 2006 available now are still not accurate enough, and consequently, residual error may become large. According to our experience, q≤180 is roughly the limitation.…”