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Casimir energy for a Regular Polygon with Dirichlet Boundaries

We study the Casimir energy of a scalar field for a regular polygon with N sides. The scalar field obeys Dirichlet boundary conditions at the perimeter of the polygon. The polygon eigenvalues $λ_N$ are expressed in terms of the Dirichlet circle eigenvalues $λ_C$ as an expansion in $\frac{1}{N}$ of the form, $λ_N = λ_C (1 + \frac{4ζ(2)}{N^2} + \frac{4ζ(3)}{N^3} + \frac{28ζ(4)}{N^4}+...)$. A comparison follows between the Casimir energy on the polygon with N=4 found with our method and the Casimir energy of the scalar field on a square. We generalize the result to spaces of the form $R^d\times P_N$, with $P_N$ a N-polygon. By the same token, we find the electric field energy for a "cylinder" of infinite length with polygonal section. With the method we use and in view of the results, it stands to reason to assume that the Casimir energy of $D$-balls has the same sign with the Casimir energy of regular shapes homeomorphic to the $D$-ball. We sum up and discuss our results at the end of the article.