elliptic function

Elliptic functions are a class of functions that generalize the familiar trigonometric functions like sine and cosine. Their study originated in the 18th century with elliptic integrals, which are integrals of rational functions involving the square root of a polynomial of degree three or four. These integrals arise naturally, for example, in calculating the length of an arc of an ellipse.1

The field was transformed in the late 1820s by mathematicians Niels Henrik Abel and Carl Jacobi. They showed that by allowing the variable to be complex and inverting the problem, a new theory emerged. These new functions, the elliptic functions, possess a property of periodicity, similar to trigonometric functions, but with two distinct periods instead of one. The theory of elliptic functions became a foundational example for the study of functions of a complex variable.1