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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] 

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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 

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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 

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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 

