# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 41 (2012) pp.135-155

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### Abstract

We consider the limit of the iteration of a map z ↦ m(z) from a complex domain D to D. For two kinds of maps m, we show that each iteration mn(z) of m(z) converges for any z ∈ D as n → ∞ and that this limit is expressed by the hypergeometric function. These are analogs of the expression of the arithmetic-geometric mean by the Gauss hypergeometric function.

MSC(Primary) 30D05 33C05, 33C65 limit of iteration, hypergeometric function