Figure 6

Boundary behavior along ${\mathit{\partial }}_{\mathbf{-}}\mathcal{A}$ (a) and ${\mathit{\partial }}_{\mathbf{+}}\mathcal{A}$ (b) of the partial sum ${\mathit{U}}_{\mathit{N}}\mathbf{(}\mathit{\varrho }\mathbf{,}\mathit{\vartheta }\mathbf{)}$ of order $\mathit{N}\mathbf{=}\mathbf{10}$ representing the solution of the Robin problem for the Helmholtz equation in the supershaped annulus described by the Gielis formula with parameters ${\mathit{k}}_{\mathit{x}}^{\mathbf{-}}\mathbf{=}{\mathit{k}}_{\mathit{y}}^{\mathbf{-}}\mathbf{=}\mathbf{2}$ , ${\mathit{d}}_{\mathit{x}}^{\mathbf{-}}\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{2}$ , ${\mathit{d}}_{\mathit{y}}^{\mathbf{-}}\mathbf{=}\mathbf{3}\mathbf{/}\mathbf{4}$ , ${\mathit{\nu }}_{\mathit{x}}^{\mathbf{-}}\mathbf{=}\mathbf{2}$ , ${\mathit{\nu }}_{\mathit{y}}^{\mathbf{-}}\mathbf{=}{\mathit{\nu }}_{\mathbf{0}}^{\mathbf{-}}\mathbf{=}\mathbf{3}$ , ${\mathit{k}}_{\mathit{x}}^{\mathbf{+}}\mathbf{=}{\mathit{k}}_{\mathit{y}}^{\mathbf{+}}\mathbf{=}{\mathit{k}}^{\mathbf{+}}\mathbf{=}\mathbf{5}$ , ${\mathit{d}}_{\mathit{x}}^{\mathbf{+}}\mathbf{=}{\mathit{d}}_{\mathit{y}}^{\mathbf{+}}\mathbf{=}\mathbf{9}\mathbf{/}\mathbf{4}$ , and ${\mathit{\nu }}_{\mathit{x}}^{\mathbf{+}}\mathbf{=}{\mathit{\nu }}_{\mathit{y}}^{\mathbf{+}}\mathbf{=}\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{-}{\mathit{\nu }}_{\mathbf{0}}^{\mathbf{+}}{\mathbf{log}}_{\mathbf{2}}\mathbf{cos}\frac{\mathit{\pi }}{{\mathit{k}}^{\mathbf{+}}}\mathbf{)}$ , ${\mathit{\nu }}_{\mathbf{0}}^{\mathbf{+}}\mathbf{\to }\mathbf{+}\mathbf{\infty }$ .