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