Elliptic function

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. 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. Formally, an elliptic function is a function f meromorphic on ℂ for which there exist two non-zero complex numbers ω1 and ω2 with ω1/ω2 ∉ ℝ, such that f(z) = f(z + ω1) and f(z) = f(z + ω2) for all z ∈ ℂ. Denoting the 'lattice of periods' by Λ = {mω1 + nω2 | m, n ∈ ℤ}, it follows that f(z) = f(z + ω) for all ω ∈ Λ. There are two families of 'canonical' elliptic functions: those of Jacobi and those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them. With the definition of elliptic functions given above (which is due to Weierstrass) the Weierstrass elliptic function ℘(z) is constructed in the most obvious way: given a lattice Λ as above, put This function is clearly invariant with respect to the transformation z ↦ z + ω for any ω ∈ Λ. The addition of the −1/ω2 terms is necessary to make the sum converge. The technical condition to ensure that an infinite sum such as this converges to a meromorphic function is that on any compact set, after omitting the finitely many terms having poles in that set, the remaining series converges normally. On any compact disk defined by |z| ≤ R, and for any |ω| > 2R, one has