What is "elliptic function"

Elliptic \El*lip"tic\, Elliptical \El*lip"tic*al\, a. [Gr. ?: cf. F. elliptique. See Ellipsis.]

1. Of or pertaining to an ellipse; having the form of an ellipse; oblong, with rounded ends.

   The planets move in elliptic orbits.
   --Cheyne.

   The billiard sharp who any one catches, His doom's extremely hard He's made to dwell In a dungeon cell On a spot that's always barred. And there he plays extravagant matches In fitless finger-stalls On a cloth untrue With a twisted cue And elliptical billiard balls!
   --Gilbert and Sullivan (The Mikado: The More Humane Mikado Song)

2. Having a part omitted; as, an elliptical phrase.

3. leaving out information essential to comprehension; so concise as to be difficult to understand; obscure or ambiguous; -- of speech or writing; as, an elliptical comment.

   Elliptic chuck. See under Chuck.

   Elliptic compasses, an instrument arranged for drawing ellipses.

   Elliptic function. (Math.) See Function.

   Elliptic integral. (Math.) See Integral.

   Elliptic polarization. See under Polarization.

n. (context mathematics English) Any function of a complex variable which is periodic in two directions

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.