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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 Fagnano and Leonhard Euler (c. 1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse .
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate (,,): first, define =, = and =. Then iterate the series
The incomplete elliptic integral of the first kind is defined as, (,) = (),the second kind as (,) = (),and the third kind as (,,) = ( ()) ().The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign.
The fundamental rectangle in the complex plane of . There are twelve Jacobi elliptic functions denoted by (,), where and are any of the letters , , , and . (Functions of the form (,) are trivially set to unity for notational completeness.) is the argument, and is the parameter, both of which may be complex.
Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss .
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass . This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p .
An elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero: see elliptic integral for the origin of the term. However, there is a natural representation of real elliptic curves with shape invariant j ≥ 1 as ellipses in the hyperbolic plane H 2 {\displaystyle \mathbb {H} ^{2}} .
The relation to elliptic integrals has mainly a historical background. Elliptic integrals had been studied by Legendre, whose work was taken on by Niels Henrik Abel and Carl Gustav Jacobi. Abel discovered elliptic functions by taking the inverse function of the elliptic integral function