Arithmetic–geometric mean

In mathematics, the arithmetic–geometric mean (AGM) of two positive real numbers $x$ and $y$ is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some mathematical constants, in particular, computing $π$ .

The AGM is defined as the limit of the interdependent sequences $a_{i}$ and $g_{i}$ :

{\begin{aligned}a_{0}&=x,\\g_{0}&=y\\a_{n+1}&={\tfrac {1}{2}}(a_{n}+g_{n}),\\g_{n+1}&={\sqrt {a_{n}g_{n}}}\,.\end{aligned}}

These two sequences converge to the same number, the arithmetic–geometric mean of $x$ and $y$ ; it is denoted by $M (x, y)$ , or sometimes by $agm(x, y)$ or $AGM(x, y)$ .

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.

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