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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 .
Legendre subsequently studied elliptic integrals and called them elliptic functions. Legendre introduced a three-fold classification –three kinds– which was a crucial simplification of the rather complicated theory at that time.
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.
where K and K′ are the complete elliptic integrals of the first kind for values satisfying k 2 + k′ 2 = 1, and E and E′ are the complete elliptic integrals of the second kind. This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a ...
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.
B. C. Carlson, John L. Gustafson 'Asymptotic approximations for symmetric elliptic integrals' 1993 arXiv; B. C. Carlson 'Numerical Computation of Real Or Complex Elliptic Integrals' 1994 arXiv; B. C. Carlson 'Elliptic Integrals:Symmetric Integrals' in Chap. 19 of Digital Library of Mathematical Functions. Release date 2010-05-07.
So the sine function is an inverse function of an integral function. [3] Elliptic functions are the inverse functions of elliptic integrals. In particular, ...
In mathematics Abel elliptic functions are a special kind of elliptic functions, that were established by the Norwegian mathematician Niels Henrik Abel. He published his paper "Recherches sur les Fonctions elliptiques" in Crelle's Journal in 1827. [1] It was the first work on elliptic functions that was actually published. [2]