THE WEIERSTRASS p-FUNCTION AS A DISTRIBUTION ON A COMPLEX TORUS, AND ITS FOURIER SERIES

摘要

We treat theWeierstrass p function associated to a lattice Lambda subset of C as a principal value distribution on the torus C/Lambda and compute its Fourier coefficients. The computation of these coefficients for nonzero frequencies is straightforward, but quite pretty. The "constant term" is more mysterious. It leads to a non-absolutely convergent doubly infinite series, which we denote sigma(1). This can be regarded as a version of an Eisenstein series, though as we discuss in 4 it differs from the "Eisenstein summation" of the series, as treated in [4]. Material from 3 on the Fourier series of elliptic functions arising from the Weierstrass zeta function leads to a formula connecting sigma(1) with the Eisenstein series treated in [4], and thereby yields a rapidly convergent approximation to sigma(1).