# elliptic

## Contents

## English[edit]

### Etymology[edit]

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*) be expressed as

_{yx}*Au*+ 2

_{xx}*Bu*+

_{xy}*Cu*+ (lower order terms) = 0. Making an analogy with the (slightly adapted) equation for a conic section (see Conic section#General Cartesian form),

_{yy}*Ax*+ 2

^{2}*Bxy*+

*Cy*+ ... = 0, a classification is made depending on the value of the discriminant

^{2}*D*=

*B*-

^{2}*AC*. The PDE is said to be

*if*

**elliptic***D*< 0,

*parabolic*if

*D*= 0 and

*hyperbolic*if

*D*> 0. (See

**Partial differential equation#Linear equations of second order**on Wikipedia.Wikipedia:Partial differential equation#Linear equations of second order)

### Adjective[edit]

**elliptic** (*not comparable*)

- (geometry) Of or pertaining to an ellipse.
**1995**, Patrick J. Roache,, page 1,**Elliptic**Marching Methods and Domain Decomposition- In this chapter, the history of solving
**elliptic**problems by direct marching methods is reviewed.

- In this chapter, the history of solving

- (mathematics) Of or pertaining to a broad field of mathematics that originates from the problem of calculating arc lengths of an ellipse.
- (mathematics, in combination, of certain functions, equations and operators) That has coefficients satisfying a condition analogous to the condition for the general equation for a conic section to be of an ellipse.

#### Synonyms[edit]

- (of or pertaining to an ellipse): elliptical

#### Coordinate terms[edit]

- (whose coefficients satisfy a condition): hyperbolic, parabolic