Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean of two positive real numbers  and  is the mutual limit of a sequence of arithmetic means and a sequence of geometric means:

Begin the sequences with x and y:

Then define the two interdependent sequences  and  as

These two sequences converge to the same number, the arithmetic–geometric mean of  and ; it is denoted by , or sometimes by  or .

The arithmetic–geometric mean is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing .

The arithmetic–geometric mean can be extended to complex numbers and when the branches of the square root are allowed to be taken inconsistently, it is, in general, a multivalued function.

Example
To find the arithmetic–geometric mean of  and , iterate as follows:

The first five iterations give the following values:

The number of digits in which  and  agree (underlined) approximately doubles with each iteration. The arithmetic–geometric mean of 24 and 6 is the common limit of these two sequences, which is approximately .

History 
The first algorithm based on this sequence pair appeared in the works of Lagrange. Its properties were further analyzed by Gauss.

Properties
The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means). As a consequence, for ,  is an increasing sequence,  is a decreasing sequence, and . These are strict inequalities if .

 is thus a number between the geometric and arithmetic mean of  and ; it is also between  and .

If , then .

There is an integral-form expression for :

where  is the complete elliptic integral of the first kind:

Indeed, since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals via this formula. In engineering, it is used for instance in elliptic filter design.

The arithmetic–geometric mean is connected to the Jacobi theta function  by

which upon setting  gives

Related concepts 
The reciprocal of the arithmetic–geometric mean of 1 and the square root of 2 is called Gauss's constant, after Carl Friedrich Gauss.

In 1799, Gauss proved that

where  is the lemniscate constant.

In 1941,  (and hence ) was proven transcendental by Theodor Schneider. The set  is algebraically independent over , but the set  (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . In fact,

The geometric–harmonic mean can be calculated by an analogous method, using sequences of geometric and harmonic means. One finds that .
The arithmetic–harmonic mean can be similarly defined, but takes the same value as the geometric mean (see section "Calculation" there).

The arithmetic–geometric mean can be used to compute – among others – logarithms,  complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.

Proof of existence
From the inequality of arithmetic and geometric means we can conclude that:

and thus

that is, the sequence  is nondecreasing.

Furthermore, it is easy to see that it is also bounded above by the larger of  and  (which follows from the fact that both the arithmetic and geometric means of two numbers lie between them). Thus, by the monotone convergence theorem, the sequence is convergent, so there exists a  such that:

However, we can also see that:

and so:

Q.E.D.

Proof of the integral-form expression
This proof is given by Gauss.
Let

Changing the variable of integration to , where

gives

Thus, we have

The last equality comes from observing that .

Finally, we obtain the desired result

Applications

The number π
For example, according to the Gauss–Legendre algorithm:

where

with  and , which can be computed without loss of precision using

Complete elliptic integral K(sinα)
Taking  and  yields the AGM

where  is a complete elliptic integral of the first kind:

That is to say that this quarter period may be efficiently computed through the AGM,

Other applications
Using this property of the AGM along with the ascending transformations of John Landen, Richard P. Brent suggested the first AGM algorithms for the fast evaluation of elementary transcendental functions (, , ). Subsequently, many authors went on to study the use of the AGM algorithms.

See also
 Landen's transformation
 Gauss–Legendre algorithm
 Generalized mean

References

Notes

Citations

Sources

 
 
 

Means
Special functions
Elliptic functions
Articles containing proofs