Mapping diagram for Moebius Transformation: roots and poles, lines and circles
Moebius Functions
Moebius or linear fractional transformations are functions , with and 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.) If , is linear and therefore the model of a holomorphic (analytic) function, and if , has a pole at . 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.