0.55 0.60 0.65 0.70 0.75 GLYPH<Minus> 0.20 GLYPH<Minus> 0.15 GLYPH<Minus> 0.10 GLYPH<Minus> 0.05 0.05 0.10 FIG. 1: Plot of 1 − α$_{m}$ ( q ) / 2 q versus c = q/m $^{2}$. While the characteristic numbers α$_{m}$ ( q ) and β$_{m}$ ( q ) for the ce$_{m}$ and se$_{m}$ modes are not explicitly known for large q ∼ m $^{2}$, which would be the case for an outstretched ellipse with b/a ≪ 1, two asymptotic formulas exist [16] α$_{m}$ = β$_{m}$ m $^{2}$≫ q = m 2 + 1 2 q 2 m 2 + 5 32 q 4 m 6 + 9 64 q 6 m 10 + 1469 8192 q 8 m 14 + 4471 16384 q 10 m 18 + . . . (C9a) α$_{m}$ = β$_{m}$ m $^{2}$≪ q = − 2 q + 4 mq 1 / 2 − 1 2 m 2 − 1 2 4 m 3 q 1 / 2 − 5 2 8 m 4 q − 33 2 12 m 5 q 3 / 2 − 63 2 14 m 6 q 2 − 527 2 18 m 7 q 5 / 2 − . . . (C9b) α$_{m}$ ( cm $^{2}$) = β$_{m}$ ( cm $^{2}$) = 2 cm 2 = 2 q . (C10) τ$_{0}$ = Z$_{m,n}$ , (C11) 3 Some results concerning the Mathieu characteristic numbers for large q ∼ m 2 can be found in Ref. [17]. 17 where Z$_{m,n}$ are the corresponding zeros. This determines the eigenvalues of the Laplacian with m and n playing the role of angular and radial quantum numbers, respectively. For They are obtained numerically with Mathematica, but are derivable from the expansions (C9a) and (C9b), correspondingly. As we see from Fig. 1, the difference 2 q − α$_{m}$ ( q ) is positive for c > 0 . 64, where the expansion (C9b) is applicable. The boundary condition requires Ce$_{m}$ ( τ$_{0}$ ) = 0 and Se$_{m}$ ( τ$_{0}$ ) = 0, which implies These expansions are not formally applicable for c = q/m 2 ∼ 1, 3 but the first formula works numerically for c ≲ 0 . 7, while second formula works numerically for c ≳ 0 . 4 − 0 . 5, so there is an overlap. Special values of c are c = 0 . 61 and c = 0 . 64, when