Abstract
Integrals of the form \( \int {R\left[ {t,\sqrt {{P\left( t \right)}} } \right]} dt \) , where P(t) is a polynomial of the third or fourth degree and R is a rational function, have the simplest algebraic integrands that can lead to nonelementary1 integrals. Equivalent integrals occur in trigonometric and other forms, in pure and applied mathematics. Such integrals are known as elliptic integrals because a special example of this type arose in the rectification of the arc of an ellipse. Although some early work on them was done by Fragnano, Euler, Lagrange and Landen, they were first treated systematically by Legendre, who showed that any elliptic integral may be made to depend on three fundamental integrals which he denoted by F(φ, k), E(φ, k) and Π(φ, n, k). These three integrals are called Legendre’s canonical elliptic integrals of the first, second and third kind respectively. Legendre’s normal forms are not the only standard forms possible, but they have retained their usefulness for over a century.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
The nonelementary character of elliptic integrals is briefly demonstrated in Integration in Finite Terms (Liouville’s Theory of Elementary Methods) by J. F. Ritt, Columbia University Press, New York, 1948, pp. 35–57.
F. Oberhettinger and W. Magnus, Anwendung der Elliptischen Funktionen in Physik und Technik. ( Grundlehren der mathematischen Wissenschaften, Band LV.) Springer-Verlag, 1949.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1971 Springer-Verlag Berlin · Heidelberg
About this chapter
Cite this chapter
Byrd, P.F., Friedman, M.D. (1971). Introduction. In: Handbook of Elliptic Integrals for Engineers and Scientists. Die Grundlehren der mathematischen Wissenschaften, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65138-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-65138-0_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-65140-3
Online ISBN: 978-3-642-65138-0
eBook Packages: Springer Book Archive