Concept# Arithmetic–geometric mean

Summary

In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means:
Begin the sequences with x and y:
\begin{align}
a_0 &= x,\
g_0 &= y.
\end{align}
Then define the two interdependent sequences (an) and (gn) as
\begin{align}
a_{n+1} &= \tfrac12(a_n + g_n),\
g_{n+1} &= \sqrt{a_n g_n}, .
\end{align}
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 is used in fast algorithms for exponential and trigonometric functions, as well as some mathematical constants, in particular, computing π.
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 functio

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