The gamma group is the set of all transformations w of the form

where a, b, c, and d are integers and .

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

1. f is meromorphic in the upper half-plane .

2. for all , where .

3. f(t) tends to a limit (possibly infinite in the sense that ) as t 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 f is meromorphic in , this condition is automatically satisfied at and and need be checked only at .

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acobi Theta Functions, Klein's Absolute Invariant, Lambda Group