Elliptic functions

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel.

Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms.

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