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.
Contents

Definition 1

Properties 2

Differential equation 3

Conditions for univalence 4

Conformal mapping of circular arc polygons 5

Complex structure on Teichmüller space 6

Diffeomorphism group of the circle 7

Notes 8

References 9
Definition
The Schwarzian derivative of a function f of one complex variable z is defined by

(Sf)(z) = \left( \frac{f''(z)}{f'(z)}\right)'  \frac{1}{2}\left({f''(z)\over f'(z)}\right)^2 = \frac{f'''(z)}{f'(z)}\frac{3}{2}\left({f''(z)\over f'(z)}\right)^2.
The alternative notation

\{f,z\} = (Sf)(z)
is frequently used.
Properties
The Schwarzian derivative of any fractional linear transformation

g(z) = \frac{az + b}{cz + d}
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

(S(f \circ g))(z) = (Sf)(g(z)) \cdot g'(z)^2.
More generally, for any sufficiently differentiable functions f and g

S(f \circ g) = \left( S(f)\circ g\right ) \cdot(g')^2+S(g).
This makes the Schwarzian derivative an important tool in onedimensional dynamics ^{[1]} since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.
Introducing the function of two complex variables^{[2]}

F(z,w)= \log \left ( \frac{f(z)f(w)}{zw} \right ),
its second mixed partial derivative is given by

\frac{\partial^2 F(z,w)}{\partial z \, \partial w} = {f^\prime(z)f^\prime(w)\over(f(z)f(w))^2}{1\over(zw)^2},
and the Schwarzian derivative is given by the formula:

(Sf)(z)= \left. 6 \cdot {\partial^2 F(z,w)\over \partial z \partial w}\right\vert_{z=w}.
The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has

(Sw)(v) = \left(\frac{dw}{dv}\right)^2 (Sv)(w)
which follows from the inverse function theorem, namely that v'(w)=1/w'.
Differential equation
The Schwarzian derivative has a fundamental relation with a secondorder linear ordinary differential equation in the complex plane.^{[3]} Let f_1(z) and f_2(z) be two linearly independent holomorphic solutions of

\frac{d^2f}{dz^2}+ Q(z) f(z)=0.
Then the ratio g(z)=f_1(z)/f_2(z) satisfies

(Sg)(z) = 2Q(z)
over the domain on which f_1(z) and f_2(z) are defined, and f_2(z) \ne 0. The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions f_1 and f_2 can be found, and furthermore, these are unique up to a common scale factor.
When a linear secondorder ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Qvalue 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 f is a holomorphic function on the unit disc, D, then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for f to be univalent is^{[4]}

S(f) \le 6(1z^2)^{2}.
Conversely if f(z) is a holomorphic function on D satisfying

S(f)(z) \le 2(1z^2)^{2},
then Nehari proved that f is univalent.^{[5]}
In particular a sufficient condition for univalence is^{[6]}

S(f)\le 2.
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 halfplane 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 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.^{[7]}^{[8]}^{[9]}
Let Δ be a circular arc polygon with angles πα_{1}, ..., πα_{n} in clockwise order. Let f : H → Δ be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points a_{1}, ..., a_{n} on the real axis. Then p(x) = S(f)(x) is realvalued for x real and not one of the points. By the Schwarz reflection principle p(x) extends to a rational function on the complex plane with a double pole at a_{i}:

p(z)=\sum_{i=1}^n \frac{(1\alpha_i^2)}{2(za_i)^2} + \frac{\beta_i}{za_i}.
The real numbers β_{i} are called accessory parameters. They are subject to 3 linear constraints:

\sum \beta_i=0

\sum 2a_i \beta_i + \left ( 1\alpha_i^2 \right ) =0

\sum a_i^2 \beta_i + a_i \left ( 1\alpha_i^2 \right ) =0
which correspond to the vanishing of the coefficients of z^{1}, z^{2} and z^{3} in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as

f(z) = {u_1(z)\over u_2(z)},
where u_1(z) and u_2(z) are linearly independent holomorphic solutions of the linear second order ordinary differential equation

u^{\prime\prime}(z) + \tfrac{1}{2} p(z)u(z)=0.
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

a(z) U^{\prime\prime}(z) + b(z) U^\prime(z) +(c(z)+\lambda)U(z)=0.
Thus q(z) u_i(z) are eigenfunctions of a SturmLiouville equation on the interval [a_i,a_{i+1}]. By the Sturm separation theorem, the nonvanishing of u_2(z) forces λ 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 D, or equivalently the upper halfplane 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 selfmap f of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere \tilde{f} onto itself. In fact \tilde{f} is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

\frac{\partial F}{\partial \overline{z}} = \mu(z) \frac{\partial F}{\partial z},
where μ is the bounded measurable function defined by

\mu(z) = \over{\partial f\over \partial z}}
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

g= S(\tilde{f}),
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.^{[10]}^{[11]}^{[12]}
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 g have the property that

g(z) dz^{2}
is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finitedimensional complex vector space of quadratic differentials on S.
Diffeomorphism group of the circle
Let F_{λ}(S^{1}) be the space of tensor densities of degree λ on S^{1}. The group of orientationpreserving diffeomorphisms of S^{1}, Diff(S^{1}), acts on F_{λ}(S^{1}) via pushforwards. If f is an element of Diff(S^{1}) then consider the mapping

f \to S(f^{1}).
In the language of group cohomology the chainlike rule above says that this mapping is a 1cocycle on Diff(S^{1})with coefficients in F_{2}(S^{1}). In fact

H^1(\text{Diff}(\mathbf{S}^1);F_2) = \mathbf{R}
and the 1cocycle generating the cohomology is f → S(f^{−1}).
There is an infinitesimal version of this result giving a 1cocycle for the Lie algebra Vect(S^{1}) of vector fields. This in turn gives the unique nontrivial central extension of Vect(S^{1}), the Virasoro algebra.
The group Diff(S^{1}) and its central extension also appear naturally in the context of Teichmüller theory and string theory.^{[13]} In fact the homeomorphisms of S^{1} induced by quasiconformal selfmaps of D are precisely the quasisymmetric homeomorphisms of S^{1}; 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 homeomorphisms QS(S^{1}) by the subgroup of Möbius transformations Moeb(S^{1}). (It can also be realized naturally as the space of quasicircles in C.) Since

\text{Moeb}(\mathbf{S}^1)\subset \text{Diff}(\mathbf{S}^1) \subset \text{QS}(\mathbf{S}^1)
the homogeneous space Diff(S^{1})/Moeb(S^{1}) 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(S^{1}) can be identified with the space of Hill's operators on S^{1}

{d^2\over d\theta^2} + q(\theta),
and the coadjoint action of Diff(S^{1}) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to

{d^2\over d\theta^2} + f^\prime(\theta)^2 \,q\circ f(\theta) + \tfrac{1}{2} S(f)(\theta).
Notes

^ Weisstein, Eric W. "Schwarzian Derivative." From MathWorldA Wolfram Web Resource.

^ Schiffer 1966

^ Hille 1976

^ Lehto 1987, p. 60

^ Duren 1983

^ Lehto 1987, p. 90

^ Nehari 1953

^ von Koppenfels & Stallmann 1959

^ Klein 1922

^ Ahlfors 1966

^ Lehto 1987

^ Imayoshi & Taniguchi 1992

^ Pekonen 1995
References

, Chapter 6, "Teichmüller Spaces"

Duren, Peter L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften 259, SpringerVerlag, pp. 258–265,

, Chapter 10, "The Schwarzian".

Imayoshi, Y.; Taniguchi, M. (1992), An introduction to Teichmüller spaces, SpringerVerlag,

von Koppenfels, W.; Stallmann, F. (1959), Praxis der konformen Abbildung, Die Grundlehren der mathematischen Wissenschaften 100, SpringerVerlag, pp. 114–141 , Section 12, "Mapping of polygons with circular arcs".

, "On the theory of generalized Lamé functions".

Lehto, Otto (1987), Univalent functions and Teichmüller spaces, SpringerVerlag, pp. 50–59, 111–118, 196–205,



Ovsienko, V.; Tabachnikov, S. (2005), Projective Differential Geometry Old and New, Cambridge University Press,

Ovsienko, Valentin; Tabachnikov, Sergei (2009), "What Is . . . the Schwarzian Derivative?" (PDF), AMS Notices 56 (01): 34–36

Pekonen, Osmo (1995), "Universal Teichmüller space in geometry and physics", J. Geom. Phys. 15: 227–251,

Schiffer, Menahem (1966), "HalfOrder Differentials on Riemann Surfaces", SIAM Journal on Applied Mathematics 14: 922–934,
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