In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius 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. It is named after the German mathematician Hermann Schwarz.
Definition The Schwarzian derivative of a holomorphic function of one complex variable is defined by
The same formula also defines the Schwarzian derivative of a function of one real variable. The alternative notation
is frequently used.
Properties The Schwarzian derivative of any Möbius transformation
is zero. Conversely, the Möbius transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a Möbius transformation.Thurston, William P. "Zippers and univalent functions." The Bieberbach conjecture (West Lafayette, Ind., 1985) 21 (1986): 185-197.
If is a Möbius transformation, then the composition has the same Schwarzian derivative as ; and on the other hand, the Schwarzian derivative of is given by the chain rule
More generally, for any sufficiently differentiable functions and
When and are smooth real-valued functions, this implies that all iterations of a function with negative (or positive) Schwarzian will remain negative (resp. positive), a fact of use in the study of one-dimensional dynamics.Weisstein, Eric W. "Schwarzian Derivative." From MathWorld—A Wolfram Web Resource.
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
or more explicitly, . This follows from the chain rule above.
Geometric interpretation William Thurston interprets the Schwarzian derivative as a measure of how much a conformal map deviates from a Möbius transformation. Let be a conformal mapping in a neighborhood of Then there exists a unique Möbius transformation such that has the same 0, 1, 2-th order derivatives at
Now To explicitly solve for it suffices to solve the case of Let and solve for the that make the first three coefficients of equal to Plugging it into the fourth coefficient, we get .
Givenup to third order, this function maps the circle to the curve . This is approximately an ellipse with semiaxes . Since Möbius transformations always map circles to circles or lines, the Schwarzian derivative measures how much deviates from a Möbius transformation, in that it maps circles to non-circles.
Cross-ratio Given a differentiable function , it transforms the cross-ratio . It is invariant under the Möbius transformation:In general, if the 4 points are infinitesimally close, then it transforms according toin particular, it can be interpreted as a quadratic differential form. This immediately yields the chain rule
Differential equation Consider the linear second-order ordinary differential equation
where is a real-valued function of a real parameter . Let denote the two-dimensional space of solutions. For , let be the evaluation functional . The map gives, for each point of the domain of , a one-dimensional linear subspace of . That is, the kernel defines a mapping from the real line to the real projective line. The Schwarzian of this mapping is well-defined, and in fact is equal to .
Owing to this interpretation of the Schwarzian, if two diffeomorphisms of a common open interval into have the same Schwarzian, then they are (locally) related by an element of the general linear group acting on the two-dimensional vector space of solutions to the same differential equation, i.e., a fractional linear transformation of .
Alternatively, consider the second-order linear ordinary differential equation in the complex plane
Let and be two linearly independent holomorphic solutions. Then the ratio satisfies
over the domain on which and are defined, and The converse is also true: if such a exists, and it is holomorphic on a simply connected domain, then two solutions and can be found, and furthermore, these are unique up to a common scale factor.
When a linear second-order ordinary differential equation can be brought into the above form, the resulting is sometimes called the Q-value of the equation.
Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.
Conditions for univalence If is a holomorphic function on the unit disc, , then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for to be univalent is
Conversely if is a holomorphic function on satisfying
then Nehari proved that is univalent.
In particular a sufficient condition for univalence is
Conformal mapping of circular arc polygons
The Schwarzian derivative and associated second-order ordinary differential equation can be used to determine the unique conformal map between the upper half-plane or unit disk and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. The accessory parameters that arise as constants of integration are related to the eigenvalues of the second-order differential equation. Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation.
Let be a circular arc polygon with angles in clockwise order. Let be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points on the real axis. Then is real-valued when is real and different from all the points . By the Schwarz reflection principle extends to a rational function on the complex plane with a double pole at :
The real numbers are called accessory parameters. They are subject to three linear constraints:
which correspond to the vanishing of the coefficients of and in the expansion of around . The mapping can then be written as
where and are linearly independent holomorphic solutions of the linear second-order ordinary differential equation
There are linearly independent accessory parameters, which can be difficult to determine in practise.
For a triangle, when , there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and is the Schwarz triangle function, which can be written in terms of hypergeometric functions.
For a quadrilateral the accessory parameters depend on one independent variable . Writing for a suitable choice of , the ordinary differential equation takes the form
Thus are eigenfunctions of a Sturm–Liouville equation on the interval . By the Sturm separation theorem, the non-vanishing of forces to be the lowest eigenvalue.
Complex structure on Teichmüller space Universal Teichmüller space is defined to be the space of real analytic quasiconformal mappings of the unit disc , or equivalently the upper half-plane , onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a Möbius transformation. Identifying with the lower hemisphere of the Riemann sphere, any quasiconformal self-map of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere onto itself. In fact is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation
where μ is the bounded measurable function defined by
on the lower hemisphere, extended to 0 on the upper hemisphere.
Identifying the upper hemisphere with , Lipman Bers used the Schwarzian derivative to define a mapping
which embeds universal Teichmüller space into an open subset of the space of bounded holomorphic functions on with the uniform norm. Frederick Gehring showed in 1977 that is the interior of the closed subset of Schwarzian derivatives of univalent functions.
For a compact Riemann surface of genus greater than 1, its universal covering space is the unit disc on which its fundamental group acts by Möbius transformations. The Teichmüller space of can be identified with the subspace of the universal Teichmüller space invariant under . The holomorphic functions have the property that
is invariant under , so determine quadratic differentials on . In this way, the Teichmüller space of is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on .
Diffeomorphism group of the circle Crossed homomorphisms The transformation property
allows the Schwarzian derivative to be interpreted as a continuous 1-cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficients in the module of densities of degree 2 on the circle. Let be the space of tensor densities of degree on . The group of orientation-preserving diffeomorphisms of , acts on via pushforwards. If then consider the mapping
In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on with coefficients in . In fact
and the 1-cocycle generating the cohomology is The computation of 1-cohomology is a particular case of the more general result
Note that if is a group and a -module, then the identity defining a crossed homomorphism of into can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism of into the semidirect product such that the composition of with the projection onto is the identity map; the correspondence is by the map
The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms
in . A simple averaging argument shows that, if is a compact group and a topological vector space on which K acts continuously, then the higher cohomology groups vanish
In particular for 1-cocycles χ with
averaging over , using left invariant of the Haar measure on gives
with
Thus by averaging it may be assumed that satisfies the normalisation condition for . Note that if any element in satisfies then
But then, since is a homomorphism,
so that satisfies the equivariance condition
Thus it may be assumed that the cocycle satisfies these normalisation conditions for . The Schwarzian derivative in fact vanishes whenever is a Möbius transformation corresponding to . The other two 1-cycles discussed below vanish only on .
There is an infinitesimal version of this result giving a 1-cocycle for , the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when is a Lie group and the action of on is smooth, there is a Lie algebraic version of crossed homomorphism obtained by taking the corresponding homomorphisms of the Lie algebras (the derivatives of the homomorphisms at the identity). This also makes sense for and leads to the 1-cocycle
which satisfies the identity
In the Lie algebra case, the coboundary maps have the form
In both cases the 1-cohomology is defined as the space of crossed homomorphisms modulo coboundaries. The natural correspondence between group homomorphisms and Lie algebra homomorphisms leads to the "van Est inclusion map"
In this way the calculation can be reduced to that of Lie algebra cohomology. By continuity this reduces to the computation of crossed homomorphisms of the Witt algebra into . The normalisations conditions on the group crossed homomorphism imply the following additional conditions for :
for in .
Following the conventions of , a basis of the Witt algebra is given by
so that
A basis for the complexification of is given by