An elliptic integral is an integral with the form

Here R is a rational function of its two arguments, w, and x, and these two arguments are related to each other by these conditions:

- w
^{2}is a cubic function or quartic function in x, i.e. w^{2}= f(x) = a_{0}x^{4}+ a_{1}x^{3}+ a_{2}x^{2}+ a_{3}x^{}+ a_{4} - R(w,x) has at least one odd power of w
- w
^{2}has no repeated roots

In a way, these integrals are generalizations of inverse trigonometric functions. They provide solutions to a wider class of problems than inverse trigonometric functions do; simple problems like calculating the position of a pendulum as well as more complicated problems in electromagnetism and gravitation.

## Reducing Elliptic Integrals

As a rule, elliptic integrals can’t be written in terms of simple functions. There are some special integrals, though: the *Legendre elliptic integrals* or the *canonical elliptic integrals* of the first, second and third kinds. Every elliptic integral can be written as a sum of elementary functions and linear combinations of these.

## History

These get their name because they were first studied by mathematicians looking to calculate the arc length of an ellipse. The first recorded study of this problem was in 1655 by John Wallis and shortly after by Isaac Newton, who both published an infinite series expansion that gave the arc length of an ellipse. Later, French mathematician Adrien Marie Legendre (who lived between 1752 and 1833) spent nearly forty years researching elliptic integrals, and he was the first to classify elliptic integrals and find ways of defining them in terms of simpler functions.

## References

Elliptic Integrals, Elliptic Functions, and Theta Functions. Retrieved from http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap3.pdf on April 22, 2019

Carlson, B. C. NIST Digital Library of Mathematical Functions. Chapter 19: Elliptic Integrals. Release 1.0.22 of 2019-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. Retrieved from https://dlmf.nist.gov/19 on April 22, 2019.

Hall, L. (1995). Special Functions. Retrieved May 15, 2019 from: http://web.mst.edu/~lmhall/SPFNS/sfch3.pdf

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