Landen's transformation


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.

Statement

The incomplete elliptic integral of the first kind is
where is the modular angle. Landen's transformation states that if,,, are such that and, then
Landen's transformation can similarly be expressed in terms of the elliptic modulus and its complement.

Complete elliptic integral

In Gauss's formulation, the value of the integral
is unchanged if and are replaced by their arithmetic and geometric means respectively, that is
Therefore,
From Landen's transformation we conclude
and.

Proof

The transformation may be effected 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

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, that is
the relationship may be written as
which may be solved for the AGM of a pair of arbitrary arguments;