Editing Möbius transformation (section)

=== Composition of simple transformations ===

If <math>c \neq 0</math>, let:
* <math>f_1(z)= z+d/c \quad</math> (translation by ''d''/''c'')
* <math>f_2(z)= 1/z \quad</math> (inversion and reflection with respect to the real axis)
* <math>f_3(z)= \frac{bc-ad}{c^2} z \quad</math> (homothety and rotation)
* <math>f_4(z)= z+a/c \quad</math> (translation by ''a''/''c'')

Then these functions can be [[Function composition|composed]], showing that, if  
<math display="block"> f(z) = \frac{az+b}{cz+d}, </math>
one has
<math display="block"> f=f_4\circ f_3\circ f_2\circ f_1 . </math>
In other terms, one has
<math display="block">\frac{az+b}{cz+d} = \frac ac + \frac e{z+\frac dc},</math>
with
<math display="block">e= \frac{bc-ad}{c^2}. </math>

This decomposition makes many properties of the Möbius transformation obvious.