Editing Elliptic integral (section)

{{Short description|Special function defined by an integral}}
{{Use American English|date = January 2019}}
In [[integral calculus]], an '''elliptic integral''' is one of a number of related functions defined as the value of certain integrals, which were first studied by [[Giulio Carlo de' Toschi di Fagnano|Giulio Fagnano]] and [[Leonhard Euler]] ({{Circa|1750}}). Their name originates from their originally arising in connection with the problem of finding the [[arc length]] of an [[ellipse]].

Modern mathematics defines an "elliptic integral" as any [[function (mathematics)|function]] {{math|''f''}} which can be expressed in the form

<math display="block"> f(x) = \int_{c}^{x} R{\left({\textstyle t, \sqrt{ P(t)} }\right)} \, dt,</math>

where {{math|''R''}} is a [[rational function]] of its two arguments, {{math|''P''}} is a [[polynomial]] of degree 3 or 4 with no repeated roots, and {{math|''c''}} is a constant.

In general, integrals in this form cannot be expressed in terms of [[elementary function]]s. Exceptions to this general rule are when {{math|''P''}} has repeated roots, or when {{math|''R''(''x'', ''y'')}} contains no odd powers of {{math|''y''}} or if the integral is pseudo-elliptic. However, with the appropriate [[Integration by reduction formulae|reduction formula]], every elliptic integral can be brought into a form that involves integrals over rational functions and the three [[Legendre form|Legendre canonical form]]s, also known as the elliptic integrals of the first, second and third kind.

Besides the Legendre form given below, the elliptic integrals may also be expressed in [[Carlson symmetric form]]. Additional insight into the theory of the elliptic integral may be gained through the study of the [[Schwarz–Christoffel mapping]]. Historically, [[elliptic functions]] were discovered as inverse functions of elliptic integrals.