The Schwarzian derivative of a three times continuously differentiable function 
is
![S(f)(x)=(f^(''')(x))/(f^'(x))-3/2[(f^('')(x))/(f^'(x))]^2,](/images/equations/SchwarzianDerivative/NumberedEquation1.svg) |
(1)
|
where 
.
The same definition applies to an analytic function
of a complex variable wherever its derivative
is nonzero.
Its chain rule for function composition is
![S(f degreesg)(x)=S(f)(g(x))[g^'(x)]^2+S(g)(x),](/images/equations/SchwarzianDerivative/NumberedEquation2.svg) |
(2)
|
where the derivatives 
and 
are nonzero. Consequently, postcomposition by a linear fractional transformation 
leaves it unchanged,
 |
(3)
|
Locally, 
iff 
with 
, away from the pole of the
linear fractional transformation
(Ovsienko and Tabachnikov 2009, pp. 34-36).
If 
and 
are linearly independent solutions of the
second-order ordinary differential
equation
 |
(4)
|
then their quotient satisfies
 |
(5)
|
wherever 
(Ovsienko and Tabachnikov 2009, pp. 34-36). The nome
and elliptic lambda function provide examples
of this connection with differential equations.
In one-dimensional dynamics, a negative Schwarzian derivative is a useful hypothesis in studying period doubling
(Tabor 1989, p. 220). For example, the logistic
map 
with 
has
 |
(6)
|
for 
.
Negativity alone does not imply the universality of the Feigenbaum
constant, whose scaling class also depends on the order of the critical
point (Feigenbaum 1978, p. 25).
