Etymology

From Ancient Greek ἐλλειπτικός ‎(elleiptikós), from ἐλλείπω ‎(elleípō, “I leave out, omit”).

The mathematical theory inspired by elliptic integrals, first studied in connection with calculating the arclength of an ellipse, has developed into a broad and still active mathematical field. Elliptic functions were discovered by Neils Abel as inverse functions of the integrals, and applications were subsequently found in physics and number theory. Karl Weierstrass later found a way of expressing all elliptic functions in terms of a simple elliptic function (see Weierstrass's elliptic functions). Besides being used to evaluate integrals and explicitly solve certain differential equations, elliptic functions have deep connections with elliptic curves (algebraic geometry) and modular forms (complex analysis, with applications in number theory, algebraic topology and string theory).

The general second-order linear partial differential equation (PDE) can (assuming u_xy = u_yx) be expressed as Au_xx + 2Bu_xy + Cu_yy + (lower order terms) = 0. Making an analogy with the (slightly adapted) equation for a conic section (see Conic section#General Cartesian form), Ax² + 2Bxy + Cy² + ... = 0, a classification is made depending on the value of the discriminant D = B² - AC. The PDE is said to be elliptic if D < 0, parabolic if D = 0 and hyperbolic if D > 0. (See Partial differential equation#Linear equations of second order on Wikipedia.Wikipedia )