Schwarzian derivative

Summary

The Schwarzian derivative of a locally injective holomorphic function $f$ is the expression $f'''/f' - \frac{3}{2}(f''/f')^2$. It vanishes exactly for Möbius transformations and does not change when $f$ is composed on the left with a Möbius transformation. So it measures how far $f$ is from being a Möbius map. It appears in univalence criteria, in the theory of linear second order differential equations, and in Teichmüller theory.