Knowlpedia

Definition

Schwarzian derivative

A third-order differential invariant of a locally univalent holomorphic function.

Definition

Let ff be holomorphic and locally univalent on a , so f′≠0f'\ne0. Its Schwarzian derivative is

S(f)=f′′′f′−32(f′′f′)2.S(f) =\frac{f'''}{f'}-\frac32\left(\frac{f''}{f'}\right)^2.

It is a holomorphic function in the chosen coordinate.

Logarithmic-derivative form

Writing u=f′′/f′u=f''/f', one has

S(f)=u′−12u2.S(f)=u'-\frac12u^2.

This form displays the nonlinear correction that makes the Schwarzian obey its special .

Geometric role

The Schwarzian measures the failure of a locally univalent map to be projective-linear: its vanishing . Under changes of source coordinate it transforms as a rather than as an ordinary function. The relates it to ratios of solutions of second-order linear differential equations.

Scope

At a critical point f′=0f'=0, the displayed expression is not holomorphic and may have a pole. One can treat the Schwarzian meromorphically in broader settings, but local univalence is the clean hypothesis for this definition.

References
  1. Zeev Nehari, Conformal Mapping, Dover, 1975. Relevant: Chapter VI.
  2. R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective connections and the Schwarzian.