Fundamental Region
المؤلف:
Apostol, T. M
المصدر:
"Fundamental Region." §2.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
...
23-12-2019
1135
Fundamental Region
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](http://mathworld.wolfram.com/images/equations/FundamentalRegion/Inline19.gif)
.
REFERENCES:
Apostol, T. M. "Fundamental Region." §2.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 30-34, 1997.
Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 112-113, 1987.
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