Schwarzian derivative

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.

- 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

Last revised on November 12, 2018 at 08:46:56. See the history of this page for a list of all contributions to it.