Generalization of Weierstrassian Elliptic Functions to Rⁿ

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The Weierstrassian, ζ and σ functions are generalized to Rⁿ. The n=3 and n=4 cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations of spacetime foam and the monopole condensate, respectively. The new functions satisfy higher dimensional versions of the periodicity properties and Legendre's relations obeyed by their familiar complex counterparts. For n=4, the construction reproduces functions found earlier by Fueter using quaternionic methods. Integrating over lattice points along all directions but two, one recovers the original Weierstrassian elliptic functions.