Upper and low bounds of azimuthal numbers related to elementary wave functions of an elliptic cylinder

Numerical and analytical aspects of generating $2\pi$-periodic solutions of the angular Mathieu equation obtained for the circumferential harmonics of an elliptic cylinder and localization problem for the Mathieu eigenvalues and corresponding azimuthal numbers are considred. Those are required in usual procedure of constructing the elliptic cylinder elementary wave functions playing a very important role in mathematical physics. The Sturm–Liouville eigenvalue problem for angular Mathieu equation is reformulated as the algebraic eigenvalue problem for a infinite linear self-adjoint pentadiagonal matrix operator acting in the complex bi-infinite sequence space $l_2$. The matrix operator then can be splitted into a diagonal matrix and a infinite symmetric doubly stochastic matrix. Simple algorithms aimed at computation of the Mathieu eigenvalues and associated angular harmonics are discussed. The most symmetric forms and equations mostly known from the contemporary theory of the Mathieu equation are systematically used. Some of them are specially derived for the case and seem to be new in the theory of the angular Mathieu equation. An extension of the azimuthal numbers notion to the case of elastic and thermoelastic waves propagating in a long elliptic waveguide is proposed. Estimations of upper and low bounds for the angular Mathieu eigenvalues and azimuthal numbers are obtained by the aid of the Gerschgorin theorems and more accurate ones by the Cassini ovals technique.