The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by
(
S
f
)
(
z
)
=
(
f
″
(
z
)
f
′
(
z
)
)
′
−
1
2
(
f
″
(
z
)
f
′
(
z
)
)
2
=
f
‴
(
z
)
f
′
(
z
)
−
3
2
(
f
″
(
z
)
f
′
(
z
)
)
2
.
The same formula also defines the Schwarzian derivative of a C^{3} function of one real variable. The alternative notation
{
f
,
z
}
=
(
S
f
)
(
z
)
is frequently used.
The Schwarzian derivative of any fractional linear transformation
g
(
z
)
=
a
z
+
b
c
z
+
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
∘
g
)
)
(
z
)
=
(
S
f
)
(
g
(
z
)
)
⋅
g
′
(
z
)
2
.
More generally, for any sufficiently differentiable functions f and g
S
(
f
∘
g
)
=
(
S
(
f
)
∘
g
)
⋅
(
g
′
)
2
+
S
(
g
)
.
This makes the Schwarzian derivative an important tool in onedimensional 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
F
(
z
,
w
)
=
log
(
f
(
z
)
−
f
(
w
)
z
−
w
)
,
its second mixed partial derivative is given by
∂
2
F
(
z
,
w
)
∂
z
∂
w
=
f
′
(
z
)
f
′
(
w
)
(
f
(
z
)
−
f
(
w
)
)
2
−
1
(
z
−
w
)
2
,
and the Schwarzian derivative is given by the formula:
(
S
f
)
(
z
)
=
6
⋅
∂
2
F
(
z
,
w
)
∂
z
∂
w

z
=
w
.
The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
(
S
w
)
(
v
)
=
−
(
d
w
d
v
)
2
(
S
v
)
(
w
)
which follows from the inverse function theorem, namely that
v
′
(
w
)
=
1
/
w
′
.
The Schwarzian derivative has a fundamental relation with a secondorder linear ordinary differential equation in the complex plane. Let
f
1
(
z
)
and
f
2
(
z
)
be two linearly independent holomorphic solutions of
d
2
f
d
z
2
+
Q
(
z
)
f
(
z
)
=
0.
Then the ratio
g
(
z
)
=
f
1
(
z
)
/
f
2
(
z
)
satisfies
(
S
g
)
(
z
)
=
2
Q
(
z
)
over the domain on which
f
1
(
z
)
and
f
2
(
z
)
are defined, and
f
2
(
z
)
≠
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.
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

S
(
f
)

≤
6
(
1
−

z

2
)
−
2
.
Conversely if f(z) is a holomorphic function on D satisfying

S
(
f
)
(
z
)

≤
2
(
1
−

z

2
)
−
2
,
then Nehari proved that f is univalent.
In particular a sufficient condition for univalence is

S
(
f
)

≤
2.
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.
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
)
=
∑
i
=
1
n
(
1
−
α
i
2
)
2
(
z
−
a
i
)
2
+
β
i
z
−
a
i
.
The real numbers β_{i} are called accessory parameters. They are subject to 3 linear constraints:
∑
β
i
=
0
∑
2
a
i
β
i
+
(
1
−
α
i
2
)
=
0
∑
a
i
2
β
i
+
a
i
(
1
−
α
i
2
)
=
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
)
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
′
′
(
z
)
+
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
′
′
(
z
)
+
b
(
z
)
U
′
(
z
)
+
(
c
(
z
)
+
λ
)
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.
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
f
~
onto itself. In fact
f
~
is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation
∂
F
∂
z
¯
=
μ
(
z
)
∂
F
∂
z
,
where μ is the bounded measurable function defined by
μ
(
z
)
=
∂
f
∂
z
¯
∂
f
∂
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
(
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.
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
)
d
z
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.
The transformation property
S
(
f
∘
g
)
=
(
S
(
f
)
∘
g
)
⋅
(
g
′
)
2
+
S
(
g
)
.
allows the Schwarzian derivative to be interpreted as a continuous 1cocycle or crossed homomorphism of the diffeomorphism group of the circle with coefficents in the module of densities of degree 2 on 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
→
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
(
Diff
(
S
1
)
;
F
2
(
S
1
)
)
=
R
and the 1cocycle generating the cohomology is f → S(f^{−1}). The computation of 1cohomology is a particular case of the more general result
H
1
(
Diff
(
S
1
)
;
F
λ
(
S
1
)
)
=
R
f
o
r
λ
=
0
,
1
,
2
a
n
d
(
0
)
o
t
h
e
r
w
i
s
e
.
Note that if G is a group and M a Gmodule, then the identity defining a crossed homomorphismc of G into M can be expressed in terms of standard homomorphisms of groups: it is encoded in a homomorphism φ of G into the semidirect product
M
⋊
G
such that the composition of φ with the projection
M
⋊
G
onto G is the identity map; the correspondence is by the map C(g) = (c(g), g). The crossed homomorphisms form a vector space and containing as a subspace the coboundary crossed homomorphisms b(g) = g ⋅ m − m for m in M. A simple averaging argument shows that, if K is a compact group and V a topological vector space on which K acts continuously, then the higher cohomology groups vanish H^{m}(K, V) = (0) for m > 0. n particular for 1cocycles χ with
χ
(
x
y
)
=
χ
(
x
)
+
x
⋅
χ
(
y
)
,
averaging over y, using left invariant of the Haar measure on K gives
χ
(
x
)
=
m
−
x
⋅
m
,
with
m
=
∫
K
χ
(
y
)
d
y
.
Thus by averaging it may be assumed that c satisfies the normalisation condition c(x) = 0 for x in Rot(S^{1}). Note that if any element x in G satisifes c(x) = 0 then C(x) = (0,x). But then, since C is a homomorphism, C(xgx^{–1}) = C(x)C(g)C(x)^{–1}, so that c satisfies the equivariance condition c(xgx^{–1})=x ⋅ c(g). Thus it may be assumed that the cocycle satisifies these normalisation conditions for Rot(S^{1}). The Schwarzian derivative in fact vanishes whenever x is a Möbius transformation corresponding to SU(1,1). The other two 1cycles discussed below vanish only on Rot(S^{1}) (λ = 0, 1).
There is an infinitesimal version of this result giving a 1cocycle for Vect(S^{1}), the Lie algebra of smooth vector fields, and hence for the Witt algebra, the subalgebra of trigonometric polynomial vector fields. Indeed, when G is a Lie group and the action of G on M 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 homomotphisms at the identity). This also makes sense for Diff(S^{1}) and leads to the 1cocycle
s
(
f
d
d
θ
)
=
d
3
f
d
θ
3
(
d
θ
)
2
which satisfies the identity
s
(
[
X
,
Y
]
)
=
X
⋅
s
(
Y
)
−
Y
⋅
s
(
X
)
.
In the Lie algebra case, the coboundary maps have the form b(X) = X ⋅ m for m in M. In both cases the 1cohomology 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"
H
1
(
Diff
(
S
1
)
;
F
λ
(
S
1
)
)
↪
H
1
(
Vect
(
S
1
)
;
F
λ
(
S
1
)
)
,
In this way the calculation can be reduced to that of Lie algebra cohomology. By continuity this reduces to the computation of crossed homomorphisms φ of the Witt algebra into F_{λ}(S^{1}). The normalisations conditions on the group crossed homomorphism imply the following additional conditions for φ:
φ
(
Ad
(
x
)
X
)
=
x
⋅
φ
(
X
)
,
φ
(
d
/
d
θ
)
=
0
for x in Rot(S^{1}).
Following the conventions of Kac & Raina (1987), a basis of the Witt algebra is given by
d
n
=
i
e
i
n
θ
d
d
θ
so that [d_{m},d_{n}] = (m – n) d_{m + n}. A basis for the complexification of F_{λ}(S^{1}) is given by
v
n
=
e
i
n
θ
(
d
θ
)
λ
,
so that
d
m
⋅
v
n
=
−
(
n
+
λ
m
)
v
n
+
m
,
g
ζ
⋅
v
n
=
ζ
n
v
n
,
for g_{ζ} in Rot(S^{1}) = T. This forces φ(d_{n}) = a_{n} ⋅ v_{n} for suitable coefficients a_{n}. The crossed homomorphism condition φ([X,Y]) = Xφ(Y) – Yφ(X) gives a recurrence relation for the a_{n}:
(
m
−
n
)
a
m
+
n
=
(
m
+
λ
n
)
a
m
−
(
n
+
λ
m
)
a
n
.
The condition φ(d/dθ) = 0, implies that a_{0} = 0. From this condition and the recurrence relation, it follows that up to scalar multiples, this has a unique nonzero solution when λ equals 0, 1 or 2 and only the zero solution otherwise. The solution for λ = 1 corresponds to the group 1cocycle
φ
1
(
f
)
=
f
′
′
/
f
′
d
θ
. The solution for λ = 0 corresponds to the group 1cocycle φ_{0}(f) = log f' . The corrresponding Lie algebra 1cocycles for λ = 0, 1, 2 are given up to a scalar multiple by
φ
λ
(
F
d
d
θ
)
=
d
λ
+
1
F
d
θ
λ
+
1
(
d
θ
)
λ
.
The crossed homomorphisms in turn give rise to the central extension of Diff(S^{1}) and of its Lie algebra Vect(S^{1}), the socalled Virasoro algebra.
The group Diff(S^{1}) and its central extension also appear naturally in the context of Teichmüller theory and string theory. 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
Moeb
(
S
1
)
⊂
Diff
(
S
1
)
⊂
QS
(
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
d
θ
2
+
q
(
θ
)
,
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
d
θ
2
+
f
′
(
θ
)
2
q
∘
f
(
θ
)
+
1
2
S
(
f
)
(
θ
)
.
The Schwarzian derivative and the other 1cocycle defined on Diff(S^{1}) can be extended to biholomorphic between open sets in the complex plane. In this case the local description leads to the theory of analytic pseudogroups, formalizing the theory of infinitedimensional groups and Lie algebras first studied by Élie Cartan in the 1910s. This is related to affine and projective structures on Riemann surfaces as well as the theory of Schwarzian or projective connections, discussed by Gunning, Schiffer and Hawley.
A holomorphic pseudogroup Γ on C consists of a collection of biholomorphisms f between open sets U and V in C which contains the identity maps for each open U, which is closed under restricting to opens, which is closed under composition (when possible), which is closed under taking inverses and such that if a biholomorphisms is locally in Γ, then it too is in Γ. The pseudogroup is said to be transitive if, given z and w in C, there is a biholomorphism f in Γ such that f(z) = w. A particular case of transitive pseudogroups are those which are flat, i.e. contain all complex translations T_{b}(z) = z + b. Let G be the group, under composition, of formal power series transformations F(z) = a_{1}z + a_{2}z^{2} + .... with a_{1} ≠ 0. A holomorphic pseudogroup Γ defines a subgroup A of G, namely the subgroup defined by the Taylor series expansion about 0 (or "jet") of elements f of Γ with f(0) = 0. Conversely if Γ is flat it is uniquely determined by A: a biholomorphism f on U is contained in Γ in if and only if the power series of T_{–f(a)} ∘ f ∘ T_{a} lies in A for every a in U: in other words the formal power series for f at a is given by an element of A with z replaced by z − a; or more briefly all the jets of f lie in A.
The group G has a natural homomorphisms onto the group G_{k} of kjets obtained by taking the truncated power series taken up to the term z^{k}. This group acts faithfully on the space of polynomials of degree k (truncating terms of order higher than k). Truncations similarly define homomorphisms of G_{k} onto G_{k − 1}; the kernel consists of maps f with f(z) = z + bz^{k}, so is Abelian. Thus the group G_{k} is solvable, a fact also clear from the fact that it is in triangular form for the basis of monomials.
A flat pseudogroup Γ is said to be "defined by differential equations" if there is a finite integer k such that homomorphism of A into G_{k} is faithful and the image is a closed subgroup. The smallest such k is said to be the order of Γ. There is a complete classification of all subgroups A that arise in this way which satisfy the additional assumptions that the image of A in G_{k} is a complex subgroup and that G_{1} equals C*: this implies that the pseudogroup also contains the scaling transformations S_{a}(z) = az for a ≠ 0, i.e. contains A contains every polynomial az with a ≠ 0.
The only possibilities in this case are that k = 1 and A = {az: a ≠ 0}; or that k = 2 and A = {az/(1−bz) : a ≠ 0}. The former is the pseudogroup defined by affine subgroup of the complex Möbius group (the az + b transformations fixing ∞); the latter is the pseudogroup defined by the whole complex Möbius group.
This classification can easily be reduced to a Lie algebraic problem since the formal Lie algebra
g
of G consists of formal vector fields F(z) d/dz with F a formal power series. It contains the polynomial vectors fields with basis d_{n} = z^{n+1} d/dz (n ≥ 0), which is a subalgebra of the Witt algebra. The Lie brackets are given by [d_{m},d_{n}] = (n − m)d_{m+n}. Again these act on the space of polynomials of degree ≤ k by differentiation—it can be identified with C[[z]]/(z^{k+1})—and the images of d_{0}, ..., d_{k – 1} give a basis of the Lie algebra of G_{k}. Note that Ad(S_{a}) d_{n}= a^{–n} d_{n}. Let
a
denote the Lie algebra of A: it is isomorphic to a subalgebra of the Lie algebra of G_{k}. It contains d_{0} and is invariant under Ad(S_{a}). Since
a
is a Lie subalgebra of the Witt algebra, the only possibility is that it has basis d_{0} or basis d_{0}, d_{n} for some n ≥ 1. There are corresponding group elements of the form f(z)= z + bz^{n+1} + .... Composing this with translations yields T_{–f(ε)} ∘ f ∘ T _{ε}(z) = cz + dz^{2} + ... with c, d ≠ 0. Unless n = 2, this contradicts the form of subgroup A; so n = 2.
The Schwarzian derivative is related to the pseudogroup for the complex Möbius group. In fact if f is a biholomorphism defined on V then φ_{2}(f) = S(f) is a quadratic differential on V. If g is a bihomolorphism defined on U and g(V) ⊆ U, S(f ∘ g) and S(g) are quadratic differentials on U; moreover S(f) is a quadratic differential on V, so that g_{∗}S(f) is also a quadratic differential on U. The identity
S
(
f
∘
g
)
=
g
∗
S
(
f
)
+
S
(
g
)
is thus the analogue of a 1cocycle for the pseudogroup of biholomorphisms with coefficients in holomorphic quadratic differentials. Similarly
φ
0
(
f
)
=
log
f
′
and
φ
1
(
f
)
=
f
′
′
/
f
′
are 1cocycles for the same pseudogroup with values in holomorphic functions and holomorphic differentials. In general 1cocycle can be defined for holomorphic differentials of any order so that
φ
(
f
∘
g
)
=
g
∗
φ
(
f
)
+
φ
(
g
)
.
Applying the above identity to inclusion maps j, it follows that φ(j) = 0 ;and hence that if f_{1} is the restriction of f_{2}, so that f_{2} ∘ j = f_{1}, then φ(f_{1}) = φ (f_{2}). On the other hand taking the local holomororphic flow defined by holomorphic vector fields,—the exponential of the vector fields—the holomorphic pseudogroup of local biholomorphisms is generated by holomorphic vector fields. If the 1cocycle φ satisfies suitable continuity or analyticity conditions, it induces a 1cocycle of holomorphic vector fields, also compatible with restriction. Accordingly it defines a 1cocycle on holomorphic vector fields on C:
φ
(
[
X
,
Y
]
)
=
X
φ
(
Y
)
−
Y
φ
(
X
)
.
Restricting to the Lie algebra of polynomial vector fields with basis d_{n} = z^{n+1} d/dz (n ≥ 1), these can be determined using the same methods of Lie algebra cohomology (as in the previous section on crossed homomorphisms). There the calculation was for the whole Witt algebra acting on densities of order k, whereas here it is just for a subalgebra acting on holomorphic (or polynomial) differentials of order k. Again, assuming that φ vanishes on rotations of C, there are nonzero 1cocycles, unique up to scalar multiples. only for differentials of degree 0, 1 and 2 given by the same derivative formula
φ
k
(
p
(
z
)
d
d
z
)
=
p
(
k
+
1
)
(
z
)
(
d
z
)
k
,
where p(z) is a polynomial.
The 1cocycles define the three pseudogroups by φ_{k}(f) = 0: this gives the scaling group (k = 0); the affine group (k = 1); and the whole complex Möbius group (k = 2). So these 1cocycles are the special ordinary differential equations defining the pseudogroup. More significantly they can be used to define corresponding affine or projective structures and connections on Riemann surfaces. If Γ is a pseudogroup of smooth mappings on R^{n}, a topological space M is said to have a Γstructure if it has a collection of charts f that are homeomorphisms from open sets V_{i} in M to open sets U_{i} in R^{n} such that, for every nonempty intersecton, the natural map from f_{i} (U_{i} ∩ U_{j}) to f_{j} (U_{i} ∩ U_{j}) lies in Γ. This defines the structure of a smooth nmanifold if Γ consists of local diffeomorphims and a Riemann surface if n = 2—so that R^{2} ≡ C—and Γ consists of biholomorphisms. If Γ is the affine pseudogroup, M is said to have an affine structure; and if Γ is the Möbius pseudogroup, M is said to have a projective structure. Thus a genus one surface given as C/Λ for some lattice Λ ⊂ C has an affine structure; and a genus p > 1 surface given as the quotient of the upper half plane or unit disk by a Fuchsian group has a projective structure.
Gunning (1966) describes how this process can be reversed: for genus p > 1, the existence of a projective connection, defined using the Schwarzian derivative φ_{2} and proved using standard results on cohomology, can be used to identify the universal covering surface with the upper half plane or unit disk (a similar result holds for genus 1, using affine connections and φ_{1}).