nLab Schwarzian derivative

Definition

The Schwarzian derivative is an operator on complex functions that is invariant under fractional linear transformations:

$$(S f)(z) := \left( \frac{f''(z)}{f'(z)}\right)^' - \, \frac{1}{2} \left( \frac{f''(z)}{f'(z)}\right)^2 = \frac{f'''(z)}{f'(z)} - \frac{3}{2}\left( \frac{f''(z)}{f'(z)}\right)^2$$

In fact the Schwarzian derivative of a fractional linear transformation, considered as a function from $\mathbb{C}$ to $\mathbb{C}$, is zero.

References

• wikipedia: Schwarzian derivative
• Brad Osgood, Old and new about Schwarzian derivative, pdf
• V. Ovsienko, S. Tabachnikov, Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups, Cambridge Tracts in Mathematics 165 (2005) MR2177471; What is…the Schwarzian derivative, Notices Amer. Math. Soc. Jan. 2009, pdf
• V. Ovsienko, Lagrange schwarzian derivative and symplectic Sturm theory, Ann.Fac.Sci.Toulouse 2:1, 73–96, 1993 numdam
• MathOverflow: Is there an underlying explanation for the magical powers of the Schwarzian derivative?

category: analysis