Arithmetic-Geometric Mean

The arithmetic-geometric mean, or AGM, of two positive real numbers x and y is the common limit of two interleaved sequences, one built from repeated arithmetic means and the other from repeated geometric means: starting from a0 = x and g0 = y, each new term is a(n+1), the average of a(n) and g(n), and g(n+1), the square root of their product, and both sequences converge to the same value, written M(x, y) or AGM(x, y). The arithmetic-geometric mean is used in fast algorithms for computing exponential and trigonometric functions and other special functions, and for computing mathematical constants, most notably pi, and it extends to complex numbers and to a multivalued function when different branches of the square root are allowed. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/