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| In [[mathematics]], the '''Schwarzian derivative''', named after the German mathematician [[Hermann Schwarz]], is a certain operator that is invariant under all [[linear fractional transformation]]s. 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.
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| ==Definition==
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| The Schwarzian derivative of a function of one [[complex variable]] ''ƒ'' is defined by
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| :<math>\begin{align}
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| (Sf)(z) & = \left( \frac{f''(z)}{f'(z)}\right)' - \frac{1}{2}\left({f''(z)\over f'(z)}\right)^2 \\
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| & = \frac{f'''(z)}{f'(z)}-\frac{3}{2}\left({f''(z)\over f'(z)}\right)^2.
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| \end{align}</math>
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| The alternative notation
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| :<math>\{f,z\} = (Sf)(z)</math>
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| is frequently used. | |
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| ==Properties==
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| The Schwarzian derivative of any [[fractional linear transformation]]
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| : <math>g(z) = \frac{az + b}{cz + d}</math>
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| 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.
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| If ''g'' is a fractional linear transformation, then the composition ''g'' <small>o</small> ''f'' has the same Schwarzian derivative as ''f''. On the other hand, the Schwarzian derivative of ''f'' <small>o</small> ''g'' is given by the [[chain rule]]
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| : <math>(S(f \circ g))(z) = (Sf)(g(z)) \cdot g'(z)^2.</math>
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| <!--:{{bigmath|(S(<VAR >f</VAR > ∘ <VAR >g</VAR >))(<VAR >z</VAR >) {{=}} (S<VAR >f</VAR >)(<VAR >g</VAR >(z)) ⋅ <VAR >g</VAR >′(<VAR >z</VAR >)²}}-->
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| More generally, for any sufficiently differentiable functions ''f'' and ''g''
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| : <math>S(f \circ g) = \left( S(f)\circ g\right ) \cdot(g')^2+S(g).</math>
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| <!--:{{bigmath|(S(<VAR >f</VAR > ∘ <VAR >g</VAR >))(<VAR >z</VAR >) {{=}} (S<VAR >f</VAR >)(<VAR >g</VAR >(z)) ⋅ <VAR >g</VAR >′(<VAR >z</VAR >)² + S(<VAR >g</VAR >)}}-->
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| This makes the Schwarzian derivative an important tool in one-dimensional [[Dynamical system|dynamics]] <ref>[http://mathworld.wolfram.com/SchwarzianDerivative.html Weisstein, Eric W. "Schwarzian Derivative." From MathWorld--A Wolfram Web Resource.]</ref> since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.
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| Introducing the function of two complex variables<ref>{{harvnb|Schiffer|1966}}</ref>
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| :<math>F(z,w)= \log \left ( \frac{f(z)-f(w)}{z-w} \right ),</math>
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| its second mixed partial derivative is given by
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| :<math> \frac{\partial^2 F(z,w)}{\partial z \partial w} = {f^\prime(z)f^\prime(w)\over(f(z)-f(w))^2}-{1\over(z-w)^2},</math>
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| and the Schwarzian derivative is given by the formula:
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| :<math> (Sf)(z)= \left. 6 \cdot {\partial^2 F(z,w)\over \partial z \partial w}\right\vert_{z=w}.</math>
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| The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
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| :<math>(Sw)(v) = -\left(\frac{dw}{dv}\right)^2 (Sv)(w)</math>
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| which follows from the [[inverse function theorem]], namely that <math>v'(w)=1/w'.</math>
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| ==Differential equation==
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| The Schwarzian derivative has a fundamental relation with a second-order linear [[Complex differential equation|ordinary differential equation in the complex plane]].<ref>{{harvnb|Hille|1976}}</ref> Let <math>f_1(z)</math> and <math>f_2(z)</math> be two [[Wronskian|linearly independent]] [[holomorphic]] solutions of
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| :<math>\frac{d^2f}{dz^2}+ Q(z) f(z)=0.</math>
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| Then the ratio <math>g(z)=f_1(z)/f_2(z)</math> satisfies
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| :<math>(Sg)(z) = 2Q(z)</math>
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| over the domain on which <math>f_1(z)</math> and <math>f_2(z)</math> are defined, and <math>f_2(z) \ne 0.</math> The converse is also true: if such a ''g'' exists, and it is holomorphic on a [[simply connected]] domain, then two solutions <math>f_1</math> and <math>f_2</math> can be found, and furthermore, these are unique [[up to]] a common scale factor.
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| 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.
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| 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.
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| ==Conditions for univalence==
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| If ''f'' is a [[holomorphic function]] on the unit disc, '''D''', then W. Kraus (1932) and [[Zeev Nehari|Nehari]] (1949) proved that a ''necessary condition'' for ''f'' to be [[univalent function|univalent]] is<ref>{{harvnb|Lehto|1987|p=60}}</ref>
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| :<math>|S(f)| \le 6.</math> | |
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| Conversely if ''f''(''z'') is a holomorphic function on '''D''' satisfying
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| :<math> |S(f)(z)| \le 2(1-|z|^2)^{-2},</math>
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| then Nehari proved that ''f'' is univalent.<ref>{{harvnb|Duren|1983}}</ref>
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| In particular a ''sufficient condition'' for univalence is<ref>{{harvnb|Lehto|1987|p=90}}</ref>
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| :<math> |S(f)|\le 2.</math> | |
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| ==Conformal mapping of circular arc polygons==
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| 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 [[Spectral theory of ordinary differential equations|eigenvalues]] of the second order differential equation. Already in 1890 [[Felix Klein]] had studied the case of quadrilaterals in terms of the [[Lamé function|Lamé differential equation]].<ref>{{harvnb|Nehari|1953}}</ref><ref>{{harvnb|von Koppenfels|Stallmann|1959}}</ref><ref>{{harvnb|Klein|1922}}</ref>
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| Let Δ be a circular arc polygon with angles πα<sub>1</sub>, ..., πα<sub>''n''</sub> 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''<sub>1</sub>, ..., ''a<sub>n</sub>'' 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 ''a<sub>i</sub>'':
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| :<math> p(z)=\sum_{i=1}^n \frac{(1-\alpha_i^2)}{2(z-a_i)^2} + \frac{\beta_i}{z-a_i}.</math>
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| The real numbers β<sub>''i''</sub> are called ''accessory parameters''. They are subject to ''3'' linear constraints:
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| :<math>\sum \beta_i=0</math>
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| :<math> \sum 2a_i \beta_i + \left ( 1-\alpha_i^2 \right ) =0</math>
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| :<math> \sum a_i^2 \beta_i + a_i \left ( 1-\alpha_i^2 \right ) =0</math>
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| which correspond to the vanishing of the coefficients of <math> z^{-1}, z^{-2}</math> and <math>z^{-3}</math> in the expansion of ''p''(''z'') around ''z'' = ∞. The mapping ''f''(''z'') can then be written as
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| :<math> f(z) = {u_1(z)\over u_2(z)},</math>
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| where <math>u_1(z)</math> and <math>u_2(z)</math> are linearly independent holomorphic solutions of the linear second order ordinary differential equation
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| :<math> u^{\prime\prime}(z) + \tfrac{1}{2} p(z)u(z)=0.</math>
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| There are ''n''−3 linearly independent accessory parameters, which can be difficult to determine in practise.
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| 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 function]]s.
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| 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
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| :<math> a(z) U^{\prime\prime}(z) + b(z) U^\prime(z) +(c(z)+\lambda)U(z)=0.</math>
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| Thus <math>q(z) u_i(z)</math> are eigenfunctions of a [[Sturm-Liouville equation]] on the interval <math>[a_i,a_{i+1}]</math>. By the [[Sturm separation theorem]], the non-vanishing of <math>u_2(z)</math> forces λ to be the lowest eigenvalue.
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| ==Complex structure on Teichmüller space==
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| [[Universal Teichmüller space]] 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 ''f'' of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere <math>\tilde{f}</math> onto itself. In fact <math>\tilde{f}</math> is determined as the restriction to the upper hemisphere of the solution of the [[Beltrami differential equation]]
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| :<math> \frac{\partial F}{\partial \overline{z}} = \mu(z) \frac{\partial F}{\partial z},</math>
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| where μ is the bounded measurable function defined by
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| :<math>\mu(z) = {{\partial f\over \partial \overline{z}}\over{\partial f\over \partial z}}</math>
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| on the lower hemisphere, extended to 0 on the upper hemisphere.
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| Identifying the upper hemisphere with '''D''', [[Lipman Bers]] used the Schwarzian derivative to define a [[Bers embedding|mapping]]
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| :<math> g= S(\tilde{f}),</math>
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| 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.<ref>{{harvnb|Ahlfors|1966}}</ref><ref>{{harvnb|Lehto|1987}}</ref><ref>{{harvnb|Imayoshi|Taniguchi|1992}}</ref>
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| 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
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| :<math>g(z) dz^{2}</math> | |
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| 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''.
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| ==Diffeomorphism group of the circle==
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| Let ''F''<sub>λ</sub>('''S'''<sup>1</sup>) be the space of [[tensor density|tensor densities]] of degree λ on '''S'''<sup>1</sup>. The group of orientation-preserving diffeomorphisms of '''S'''<sup>1</sup>, Diff('''S'''<sup>1</sup>), acts on ''F''<sub>λ</sub>('''S'''<sup>1</sup>) via [[pushforward (differential)|pushforwards]]. If ''f'' is an element of Diff('''S'''<sup>1</sup>) then consider the mapping
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| :<math>f \to S(f^{-1}).</math>
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| In the language of [[group cohomology]] the chain-like rule above says that this mapping is a 1-cocycle on Diff('''S'''<sup>1</sup>)with coefficients in ''F''<sub>2</sub>('''S'''<sup>1</sup>). In fact
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| :<math>H^1(\text{Diff}(\mathbf{S}^1);F_2) = \mathbf{R}</math>
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| and the 1-cocycle generating the cohomology is ''f'' → ''S''(''f''<sup>−1</sup>).
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| There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra Vect('''S'''<sup>1</sup>) of [[vector field]]s. This in turn gives the unique non-trivial central extension of Vect('''S'''<sup>1</sup>), the [[Virasoro algebra]].
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| The group Diff('''S'''<sup>1</sup>) and its central extension also appear naturally in the context of Teichmüller theory and [[string theory]].<ref>{{harvnb|Pekonen|1995}}</ref> In fact the homeomorphisms of '''S'''<sup>1</sup> induced by quasiconformal self-maps of '''D''' are precisely the [[quasisymmetric map|quasisymmetric homeomorphisms]] of '''S'''<sup>1</sup>; 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('''S'''<sup>1</sup>) by the subgroup of Möbius transformations Moeb('''S'''<sup>1</sup>). (It can also be realized naturally as the space of [[quasicircle]]s in '''C'''.) Since
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| :<math>\text{Moeb}(\mathbf{S}^1)\subset \text{Diff}(\mathbf{S}^1) \subset \text{QS}(\mathbf{S}^1)</math>
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| the [[homogeneous space]] Diff('''S'''<sup>1</sup>)/Moeb('''S'''<sup>1</sup>) 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'''<sup>1</sup>) can be identified with the space of [[Hill's equation|Hill's operators]] on '''S'''<sup>1</sup>
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| :<math>{d^2\over d\theta^2} + q(\theta),</math>
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| and the [[coadjoint action]] of Diff('''S'''<sup>1</sup>) invokes the Schwarzian derivative. The inverse of the diffeomorphism ''f'' sends the Hill's operator to
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| :<math>{d^2\over d\theta^2} + f^\prime(\theta)^2 \,q\circ f(\theta) + \tfrac{1}{2} S(f)(\theta).</math>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966|pages=117–146}}, Chapter 6, "Teichmüller Spaces"
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| *{{citation|last=Duren|first=Peter L.|title=Univalent functions|publisher=Springer-Verlag|series=Grundlehren der Mathematischen Wissenschaften |volume=259|year= 1983|isbn= 0-387-90795-5|pages=258–265}}
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| *{{citation|first=Einar|last=Hille|authorlink=Einar Hille|title=Ordinary differential equations in the complex domain|publisher=Dover|year=1976|isbn=0-486-69620-0|pages=374–401}}, Chapter 10, "The Schwarzian".
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| *{{citation|first=Y.|last=Imayoshi|first2=M.|last2=Taniguchi|title=An introduction to Teichmüller spaces|publisher=Springer-Verlag|year=1992|isbn=4-431-70088-9}}
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| *{{citation|first=W.|last=von Koppenfels|first2=F.|last2=Stallmann|title=Praxis der konformen Abbildung|year=1959|publisher=Springer-Verlag|series=Die Grundlehren der mathematischen Wissenschaften|volume=100|pages=114–141}}, Section 12, "Mapping of polygons with circular arcs".
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| *{{citation|first=Felix|last=Klein|authorlink=Felix Klein|title=Collected works|volume=2|pages=540–549|url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN237843552|publisher=Springer-Verlag
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| |year=1922}}, "On the theory of generalized Lamé functions".
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| *{{citation|first=Otto|last=Lehto|title=Univalent functions and Teichmüller spaces|publisher=Springer-Verlag|year=1987|isbn=0-387-96310-3|pages=50–59, 111–118, 196–205}}
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| *{{citation | last1=Nehari | first1=Zeev |authorlink=Zeev Nehari| title=The Schwarzian derivative and schlicht functions | doi=10.1090/S0002-9904-1949-09241-8 | id={{MathSciNet | id = 0029999}} | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=545–551}}
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| *{{citation|first=Zeev|last=Nehari|authorlink=Zeev Nehari|title=Conformal mapping|publisher=Dover|year=1952|isbn=0-486-61137-X|pages=189–226}}
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| *{{citation|first=V. |last=Ovsienko|first2=S.|last2= Tabachnikov |title=Projective Differential Geometry Old and New|publisher= Cambridge University Press|year=2005|isbn=0-521-83186-5}}
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| * {{citation | first1 = Valentin | last1 = Ovsienko | first2 = Sergei | last2 = Tabachnikov | title = What Is . . . the Schwarzian Derivative? | journal = AMS Notices | volume = 56 | issue = 01 | pages = 34–36 | year = 2009
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| | url = http://www.ams.org/notices/200901/tx090100034p.pdf }}
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| *{{citation|last=Pekonen|first=Osmo|title=Universal Teichmüller space in geometry and physics|journal=J. Geom. Phys.|volume=15|year= 1995|pages= 227–251|
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| url=http://www.sciencedirect.com/science/article/pii/039304409400007Q|doi=10.1016/0393-0440(94)00007-Q}}
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| * {{citation|title=Half-Order Differentials on Riemann Surfaces|first=
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| Menahem|last= Schiffer|journal=SIAM Journal on Applied Mathematics|volume=14|pages=922–934|year=1966|jstor=2946143}}
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| [[Category:Projective geometry]]
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| [[Category:Modular forms]]
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| [[Category:Ordinary differential equations]]
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| [[Category:Complex analysis]]
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| [[Category:Conformal mapping]]
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