Legendre form

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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-minor 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.

[edit] 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

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

The argument $\scriptstyle{n}$ of the third kind of integral is known as the characteristic, and is the subject of a number of conflicting conventions. The characteristic may appear as either the first, second or third argument of $\scriptstyle{\Pi}$ , and furthermore it is sometimes defined with the opposite sign. Regarding the sign of $\scriptstyle{n}$ , the convention adopted here is that of Abramowitz and Stegun^[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 limit of the integrals, to $\scriptstyle{\pi / 2}$ .

The Legendre form of an elliptic curve is given by

y 2 = x (x - 1)(x - λ)

[edit] 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].

[edit] References

^ Gratton-Guinness, Ivor (1997). The Fontana History of the Mathematical Sciences. Fontana Press. pp. 308. ISBN 000686179-2.
^ Milton Abramowitz and Irene A. Stegun, (1964). "Chap. 17 Elliptic Integrals". Handbook of Mathematical Functions. Dover. pp. 589–628. ISBN 0-486-61272-4. http://www.math.sfu.ca/~cbm/aands/page_589.htm.
^ ^a ^b 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 (2 ed.). Cambridge University Press. pp. 261–271. ISBN 0-521-43108-5. http://www.nrbook.com/a/bookcpdf/c6-11.pdf.

[edit] See also

Carlson symmetric form

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[0] Gratton-Guinness, Ivor (1997). The Fontana History of the Mathematical Sciences. Fontana Press. pp. 308. ISBN 000686179-2.

[1] Milton Abramowitz and Irene A. Stegun, (1964). "Chap. 17 Elliptic Integrals". Handbook of Mathematical Functions. Dover. pp. 589–628. ISBN 0-486-61272-4. http://www.math.sfu.ca/~cbm/aands/page_589.htm.

[numerical_recipes-2] 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 (2 ed.). Cambridge University Press. pp. 261–271. ISBN 0-521-43108-5. http://www.nrbook.com/a/bookcpdf/c6-11.pdf.

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Legendre form

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Numerical Evaluation

[edit] References

[edit] See also

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