A holomorphic structure of the arithmetic-geometric mean of Gauss (Q1120708)

Gauss observed that the arithmetic-geometric mean M(a,b) is essentially an elliptic integral and that this fact is (a) very useful for numerical work, and (b) it is theoretically illuminating. The function M(a,b) is however only well-defined for positive arguments; in particular if a, b are complex, then M(a,b) can be given a meaning but it is not unique, and this non-uniqueness is closely associated with the periods of the elliptic integral. The purpose of this note is to give a monodromy- theoretic discussion of the set of possible values of M(a,b), thereby reproving some results of Cox and Geppert.