Landen

John Landen

John Landen was an English mathematician,He was born at Peakirk near Peterborough in Northamptonshire, and died at Milton in the same county...

's transformation, independently rediscovered by Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

, is a mapping of the parameters of an elliptic integral

Elliptic integral

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

, which leaves the value of the integral unchanged.

In Gauss's formulation,


is unchanged if and are replaced by their arithmetic

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 and geometric mean

Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

s respectively, that is

Proof

The transformation, may be achieved purely by integration by substitution

Integration by substitution

In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...

. It is convenient to first cast the integral in an algebraic

Algebraic function

In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...

 form by a substitution of , giving


A further substitution of gives the desired result (in the algebraic form)


This latter step is facilitated by writing the radical as


and the infinitesimal as


so that the factor of is easily 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 (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.