The group 
of all Möbius transformations of the form
 |
(1) |
where a, b, c, and d are integers with 
.
matrix
 |
(2) |
where 
.
can be expressed in the form
 |
(3) |
where
 |
 |
 |
(4) |
 |
|
 |
(5) |
although the representation is not unique (Apostol 1997, pp. 28-29).
Klein's Absolute Invariant, Möbius Transformation, Modular Group Gamma0, Modular Group Lambda, Theta Functions, Unimodular Transformation
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.