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 ![R[x+iy]](/images/equations/Gamma-ModularFunction/Inline20.svg)
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 
.
