Schwarzian derivative
Encyclopedia
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 function
s, conformal mapping and Teichmüller space
s.
The alternative notation
is frequently used.
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 is a fractional linear transformation, then the composition has the same Schwarzian derivative as . On the other hand, the Schwarzian derivative of is given by the chain rule
More generally, for any sufficiently differentiable functions and
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
. Let
and 
be two linearly independent
holomorphic solutions of
Then the ratio
satisfies
over the domain on which
and 
are defined, and 
The converse is also true: if such a g 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 Q 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.
on the unit disc |z| < 1, then W. Kraus (1932) and Nehari
(1949) proved that a necessary condition for f to be univalent
is
Conversely if f(z) is a holomorphic function on the unit disc satisfying
then Nehari proved that f is univalent.
In particular a sufficient condition for univalence is
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 Δ be a circular arc polygon with angles
in clockwise order. Let f(z) be a holomorphic map from the upper half-plane to Δ extending continuously to a map between the boundaries. Let the vertices correspond to points 
on the real axis. Then p(z)=S(f)(z) is real-valued for x real and not one of the points. By the Schwarz reflection principle
p(z) 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 3 linear constraints:
which correspond to the vanishing of the coefficients of
, 
and 
in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as
where
and 
are linearly independent holomorphic solutions of the linear second order ordinary differential equation
There are n-3 linearly independent accessory parameters, which can be difficult to determine in practise.
For a triangle, when n=3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation
and f(z) can be written in terms of hypergeometric functions.
For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z)=q(z)u(z) for a suitable choice of q(z), 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 λ to be the lowest eigenvalue.
is defined to be the space of real analytic quasiconformal mapping
s of the unit disc D, or equivalently the upper half-plane H, 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 D 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 D, Lipman Bers
used the Schwarzian derivative to define a mapping
which embeds universal Teichmüller space into an open subset U of the space of bounded holomorphic functions g on D with the uniform norm. Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions.
For a compact Riemann surface
S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space
of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions
have the property that
is invariant under Γ, so determine quadratic differential
s on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.
be the space of tensor densities
of degree
on S1. The group of orientation-preserving diffeomorphisms of S1, Diff(S1), acts on 
via pushforwards. If f is an element of Diff(S1) 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 f → S(f−1).
There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra
of vector field
s. This in turn gives the unique non-trivial central extension of
, the Virasoro algebra
.
The group Diff(S1) and its central extension also appear naturally in the context of Teichmüller theory and string theory
. In fact the homeomorphisms of the circle induced by quasiconformal self-maps of the unit disc are precisely the quasisymmetric homeomorphisms
of the circle; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homoemorphisms QS(S1) by the subgroup of Möbius transformations Moeb(S1). Since
the homogeneous space
Diff(S1) / Moeb(S1) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S1) can be identified with the space of Hill's operators on S1
and the coadjoint action of Diff(S1) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Schwarzian derivative, named after the German mathematician Hermann Schwarz
Hermann Schwarz
Karl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
, 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 function
Univalent function
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is one-to-one.- Examples:...
s, conformal mapping and Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
s.
Definition
The Schwarzian derivative of a function of one complex variable ƒ is defined byThe alternative notation
is frequently used.
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 is a fractional linear transformation, then the composition has the same Schwarzian derivative as . On the other hand, the Schwarzian derivative of is given by the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
More generally, for any sufficiently differentiable functions and
This makes the Schwarzian derivative an important tool in one-dimensional dynamics
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
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
Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
, namely that
Differential equation
The Schwarzian derivative has a fundamental relation with a second-order linear ordinary differential equation in the complex planeComplex differential equation
A complex differential equation is a differential equation whose solutions are functions of a complex variable.Constructing integrals involves choice of what path to take, which means singularities and branch points of the equation need to be studied...
. Let
and be two linearly independentWronskian
In mathematics, the Wronskian is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes be used to show that a set of solutions is linearly independent.-Definition:...
holomorphic solutions of
Then the ratio
satisfiesover the domain on which
and are defined, and The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions and can be found, and furthermore, these are unique up toUp to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
a common scale factor.
When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation.
Note that the Gaussian hypergeometric differential equation
Hypergeometric differential equation
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary 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 f(z) is a holomorphic functionHolomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
on the unit disc |z| < 1, then W. Kraus (1932) and Nehari
Zeev Nehari
Zeev Nehari was a mathematician who worked on univalent functions and on differential and integral equations...
(1949) proved that a necessary condition for f to be univalent
Univalent function
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is one-to-one.- Examples:...
is
Conversely if f(z) is a holomorphic function on the unit disc satisfying
then Nehari proved that f 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 Riemann mapping between the upper half-plane or unit circle 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 eigenvaluesSpectral theory of ordinary differential equations
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation...
of the second order differential equation. Already in 1890 Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
had studied the case of quadrilaterals in terms of the Lamé differential equation
Lamé function
In mathematics, a Lamé function is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates...
.
Let Δ be a circular arc polygon with angles
in clockwise order. Let f(z) be a holomorphic map from the upper half-plane to Δ extending continuously to a map between the boundaries. Let the vertices correspond to points on the real axis. Then p(z)=S(f)(z) is real-valued for x real and not one of the points. By the Schwarz reflection principleSchwarz reflection principle
This article is about the reflection principle in complex analysis. For the reflection principles of set theory, see Reflection principleIn mathematics, the Schwarz reflection principle is a way to extend the domain of definition of an analytic function of a complex variable F, which is defined on...
p(z) 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 3 linear constraints:
which correspond to the vanishing of the coefficients of
, and in the expansion of p(z) around z = ∞. The mapping f(z) can then be written aswhere
and are linearly independent holomorphic solutions of the linear second order ordinary differential equationThere are n-3 linearly independent accessory parameters, which can be difficult to determine in practise.
For a triangle, when n=3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation
Hypergeometric differential equation
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation...
and f(z) can be written in terms of hypergeometric functions.
For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z)=q(z)u(z) for a suitable choice of q(z), the ordinary differential equation takes the form
Thus
are eigenfunctions of a Sturm-Liouville equation on the interval . By the Sturm separation theoremSturm separation theorem
In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of homogeneous second order linear differential equations...
, the non-vanishing of
forces λ to be the lowest eigenvalue.Complex structure on Teichmüller space
Universal Teichmüller spaceUniversal Teichmüller space
In mathematical complex analysis, universal Teichmüller space T is a Teichmüller space containing the Teichmüller space T of every Fuchsian group G. It was introduced by as the set of boundary values of quasiconformal maps of the upper half-plane that fix 0, 1, and ∞....
is defined to be the space of real analytic quasiconformal mapping
Quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity....
s of the unit disc D, or equivalently the upper half-plane H, 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 D with the lower hemisphere of the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
, 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 equationwhere μ is the bounded measurable function defined by
on the lower hemisphere, extended to 0 on the upper hemisphere.
Identifying the upper hemisphere with D, Lipman Bers
Lipman Bers
Lipman Bers was an American mathematician born in Riga who created the theory of pseudoanalytic functions and worked on Riemann surfaces and Kleinian groups.-Biography:...
used the Schwarzian derivative to define a mapping
which embeds universal Teichmüller space into an open subset U of the space of bounded holomorphic functions g on D with the uniform norm. Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions.
For a compact Riemann surface
Compact Riemann surface
In mathematics, a compact Riemann surface is a complex manifold of dimension one that is a compact space. Riemann surfaces are generally classified first into the compact and the open .A compact Riemann surface C that is a...
S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions
have the property thatis invariant under Γ, so determine quadratic differential
Quadratic differential
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle.If the section is holomorphic, then the quadratic differentialis said to be holomorphic...
s on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.
Diffeomorphism group of the circle
Let be the space of tensor densitiesTensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...
of degree
on S1. The group of orientation-preserving diffeomorphisms of S1, Diff(S1), acts on via pushforwards. If f is an element of Diff(S1) then consider the mappingIn the language of group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
the chain-like rule above says that this mapping is a 1-cocycle on
with coefficients in . In factand the 1-cocycle generating the cohomology is f → S(f−1).
There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra
of vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s. This in turn gives the unique non-trivial central extension of
, the Virasoro algebraVirasoro algebra
In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
.
The group Diff(S1) and its central extension also appear naturally in the context of Teichmüller theory and string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. In fact the homeomorphisms of the circle induced by quasiconformal self-maps of the unit disc are precisely the quasisymmetric homeomorphisms
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the...
of the circle; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homoemorphisms QS(S1) by the subgroup of Möbius transformations Moeb(S1). Since
the homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
Diff(S1) / Moeb(S1) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S1) can be identified with the space of Hill's operators on S1
and the coadjoint action of Diff(S1) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to