[b][size=100][center]Moebius Functions[/center]Moebius [/size][/b][size=100]or [/size][b][size=100]linear fractional [/size][/b][size=100]transformations[/size][b][size=100] are functions [math]f(z)=\frac{az+b}{cz+d}[/math] , with [math]a,b,c,d\in\mathbb{C}[/math] and [math]ad-bc\ne0.[/math][br][/size][/b][size=100]These functions are special for complex analysis for many reasons- primarily because they are the model for all meromorphic functions (functions defined on an open set which are holomorphic on all of D except for a set of isolated points, which are poles of the function.)[br][br]If [math]c=0[/math], [math]f[/math] is linear and therefore the model of a holomorphic (analytic) function, and if [math]c\ne0[/math], [math]f[/math] has a pole at [math]z=-\frac{d}{c}[/math].[br][br]Also of special interest is the geometry of these functions. A Moebius function transforms the family of all circles and lines in the plane into itself, that is, as a geometric transformation it leaves the family of all circles and lines invariant. [br][/size]