Legendre form

In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because^[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity $\scriptstyle{k}$ (the ellipse being defined parametrically by $\scriptstyle{x = \sqrt{1 - k^{2}} \cos(t)}$, $\scriptstyle{y = \sin(t)}$).

In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.

Definition

The incomplete elliptic integral of the first kind is defined as,

$F(\phi,k) = \int_0^\phi \frac{1}{\sqrt{1 - k^2 \sin^2(t)}} dt,$

the second kind as

$E(\phi,k) = \int_0^\phi \sqrt{1 - k^2 \sin^2(t)}\,dt,$

and the third kind as

$\Pi(\phi,n,k) = \int_0^\phi \frac{1}{(1 - n \sin^2(t))\sqrt{1 - k^2 \sin^2(t)}}\,dt.$

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 argument order shown above is that of Gradshteyn and Ryzhik

^[2] as well as Numerical Recipes.^[3] The choice of sign is that of Abramowitz and Stegun^[4] as well as Gradshteyn and Ryzhik,^[2] but corresponds to the $\scriptstyle{\Pi(\phi,-n,k)}$ of Numerical Recipes.^[3]

The respective complete elliptic integrals are obtained by setting the amplitude, $\scriptstyle{\phi}$, the upper limit of the integrals, to $\scriptstyle{\pi / 2}$.

The Legendre form of an elliptic curve is given by

$y^2 = x(x - 1)(x - \lambda)$

Numerical evaluation

The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus $\scriptstyle{k}$ towards zero, while increasing the amplitude $\scriptstyle{\phi}$. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of $\scriptstyle{k}$, zero or one, the integral is readily evaluated.

Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.^[3]

References

1. Gratton-Guinness, Ivor (1997). The Fontana History of the Mathematical Sciences. Fontana Press. p. 308. ISBN 0-00-686179-2.
2. Градштейн, И. С.; Рыжик, И. М. (1971). "8.1: Special Functions: Elliptic Integrals and Functions". In Геронимус, Ю. В.; Цейтлин, М. Ю́. Tablitsy integralov, summ, rjadov i proizvedenii Таблицы интегралов, сумм, рядов и произведений [Tables of Integrals, Sums, Series, and Products] (in Russian) (5 ed.). Moscow: Nauka. LCCN 78876185.
3. William H. Press, Saul A. Teukolsky, William T. Vetterlimg and Brian P. Flannery (1992). "Chap. 6.11 Special Functions: Elliptic Integrals and Jacobian Functions". Numerical Recipes in C (PDF) (2 ed.). Cambridge University Press. pp. 261–271. ISBN 0-521-43108-5.
4. Milne-Thomson, Louis Melville (1983) [June 1964]. "Chapter 17: Elliptic Integrals". In Abramowitz, Milton; Stegun, Irene Ann. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 589, 589–628. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.

See also

• Carlson symmetric form