Abstract
We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For g =1, we give a complete description of the envelopes of holomorphy of such solutions.
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