Landen's transformation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.[1]

Statement[edit]

The incomplete elliptic integral of the first kind F is

where is the modular angle. Landen's transformation states that if , , , are such that and , then[2]

Landen's transformation can similarly be expressed in terms of the elliptic modulus and its complement .

Proof of a special case[edit]

Consider an example when the transformation does not change the value of the integral. Let

and and are replaced by their arithmetic and geometric means respectively, that is

Therefore,

From equation (aa) we conclude

and .

The same equation can be proved by integration by substitution. It is convenient to first cast the integral in an algebraic form by a substitution of , giving

A further substitution of gives the desired result

This latter step is facilitated by writing the radical as

and the infinitesimal as

so that the factor of is recognized and cancelled between the two factors.

Arithmetic-geometric mean and Legendre's first integral[edit]

If the transformation is iterated a number of times, then the parameters and converge very rapidly to a common value, even if they are initially of different orders of magnitude. The limiting value is called the arithmetic-geometric mean of and , . In the limit, the integrand becomes a constant, so that integration is trivial

The integral may also be recognized as a multiple of Legendre's complete elliptic integral of the first kind. Putting

Hence, for any , the arithmetic-geometric mean and the complete elliptic integral of the first kind are related by

By performing an inverse transformation (reverse arithmetic-geometric mean iteration), that is

the relationship may be written as

which may be solved for the AGM of a pair of arbitrary arguments;

The definition adopted here for differs from that used in the arithmetic-geometric mean article, such that here is in that article.

References[edit]

  1. ^ Gauss, C. F.; Nachlass (1876). "Arithmetisch geometrisches Mittel, Werke, Bd. 3". Königlichen Gesell. Wiss., Göttingen: 361–403. 
  2. ^ Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. 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.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.