Modular Group Gamma

The group  of all Möbius transformations of the form

(1)

where , , , and  are integers with . The group can be represented by the  matrix

(2)

where . Every  can be expressed in the form

(3)

where

			(4)
			(5)

although the representation is not unique (Apostol 1997, pp. 28-29).

REFERENCES:

Apostol, T. M. "The Modular Group and Modular Functions." Ch. 2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 17 and 26-46, 1997.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 113, 1987.