Properties
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 ∘ f has the same Schwarzian derivative as f. On the other hand, the Schwarzian derivative of f ∘ 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.
Introducing the function of two complex variables
its second mixed partial derivative is given by
and the Schwarzian derivative is given by the formula:
The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
which follows from the inverse function theorem, namely that
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