A function of the form
       az + b
f(z) = ------ ,
       cz + d
where the parameters a,b,c,d are complex numbers or real numbers or even integers (depending on the application).

There are lots of useful properties of möbius functions. Among the places where möbius functions find application are complex analysis (constraining the parameters appropriately yields the holomorphic automorphisms of the unit disk), continued fractions (where they exactly describe the operations used in constructing a continued fraction), computer arithmetic (due to their connection to continued fractions) and hyperbolic geometry of the plane (for the same reason as their use in complex analysis: they are the isometries of the hyperbolic plane when constrained appropriately).