Calculus How To

Elliptic Integral: Simple Definition. Example

Calculus Definitions >

An elliptic integral is an integral with the form

elliptic integral

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:

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.


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.


Elliptic Integrals, Elliptic Functions, and Theta Functions. Retrieved from 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 on April 22, 2019.
Hall, L. (1995). Special Functions. Retrieved May 15, 2019 from:

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