Gamma-Modular Function

The modular group Gamma is the set of all transformations  of the form

where , , , and  are integers and .

A -modular function is then defined (Borwein and Borwein 1987, p. 114) as a function  that satisfies:

1.  is meromorphic in the upper half-plane .

2.  for all , where .

3.  tends to a limit (possibly infinite in the sense that ) as  tends to the vertices of the fundamental region  where the approach is from within the fundamental region . (In the case , convergence is uniform in  as .) The vertices of the fundamental region are ,  and . Since  is meromorphic in , this condition is automatically satisfied at  and  and need be checked only at .

REFERENCES:

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.