In complex analysis, an elliptic function is a function defined on the complex plane (see Argand diagram) that is periodic in two directions. The elliptic functions can be thought of as analogs of the trigonometric functions (which have only a single period). Leading eighteenth-century mathematicians, including Leonhard Euler and Joseph Lagrange, had studied elliptic integrals, such as the integral that gives the arc length of an ellipse; however, these cannot be expressed in terms of the elementary functions (polynomials, exponentials, and trigonometric functions). It was the insight of Karl Jacobi, and also of Karl Gauss and Niels Abel, that the inverse functions of elliptic integrals are much easier to study. They turn out to be doubly periodic functions of a complex variable. While a singly periodic function like sine has a number a (specifically a = 2π) so that sin(x+a) = sin(x), a doubly periodic function f has the property that there are two numbers a, b, not rational multiples of each other, so that f(x+a) = f(x+b) = f(x). As Jacobi proved in 1834, the ratio a/b is necessarily an imaginary number.