Let 
be a subgroup of the modular
group Gamma. Then an open subset 
of the upper half-plane 
is called a fundamental region of 
if
1. No two distinct points of 
are equivalent under 
,
2. If 
,
then there is a point 
in the closure of 
such that 
is equivalent to 
under 
.
A fundamental region 
of the modular group
Gamma is given by 
such that 
and 
, illustrated above, where 
is the complex conjugate
of 
(Apostol 1997, p. 31). Borwein and Borwein (1987, p. 113) define the boundaries
of the region slightly differently by including the boundary points with ![R[tau]<=0](/images/equations/FundamentalRegion/Inline19.svg)
.
