Schwarzian derivative
In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.
The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by
The same formula also defines the Schwarzian derivative of a C3 function of one real variable. The alternative notation
is frequently used.
The Schwarzian derivative of any fractional linear transformation
is zero. Conversely, the fractional linear transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a fractional linear transformation.
If g is a fractional linear transformation, then the composition g o f has the same Schwarzian derivative as f. On the other hand, the Schwarzian derivative of f o g is given by the chain rule
More generally, for any sufficiently differentiable functions f and g
This makes the Schwarzian derivative an important tool in one-dimensional dynamics since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.
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