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 .
80-512: The Schwarzian derivative of a holomorphic function f of one complex variable z is defined by The same formula also defines the Schwarzian derivative of a C function of one real variable . The alternative notation is frequently used. The Schwarzian derivative of any Möbius transformation is zero. Conversely, the Möbius transformations are the only functions with this property. Thus,
160-410: A {\displaystyle a} in a neighbourhood of a {\displaystyle a} . In fact, f {\displaystyle f} coincides with its Taylor series at a {\displaystyle a} in any disk centred at that point and lying within the domain of the function. From an algebraic point of view,
240-402: A , {\displaystyle a,} it suffices to solve the case of z 0 = 0. {\displaystyle z_{0}=0.} Let M − 1 ( z ) = {\displaystyle M^{-1}(z)={}} ( A z + B ) / ( C z + 1 ) , {\displaystyle (Az+B)/(Cz+1),} and solve for
320-431: A domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles ), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had instead used the term synectic . Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function
400-511: A real function , except that all quantities are complex. In particular, the limit is taken as the complex number z {\displaystyle z} tends to z 0 {\displaystyle z_{0}} , and this means that the same value is obtained for any sequence of complex values for z {\displaystyle z} that tends to z 0 {\displaystyle z_{0}} . If
480-418: A broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain . That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis . Holomorphic functions are also sometimes referred to as regular functions . A holomorphic function whose domain
560-515: 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. If f is a holomorphic function on
640-412: A complex derivative in a neighbourhood is a very strong condition: It implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic ). Holomorphic functions are the central objects of study in complex analysis . Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in
720-416: 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 F λ ( S ) be the space of tensor densities of degree λ on S . The group of orientation-preserving diffeomorphisms of S , Diff( S ) , acts on F λ ( S ) via pushforwards . If f is an element of Diff( S ) then consider
800-402: A contour integral using Cauchy's differentiation formula : for any simple loop positively winding once around a {\displaystyle a} , and for infinitesimal positive loops γ {\displaystyle \gamma } around a {\displaystyle a} . In regions where the first derivative
880-492: A domain U {\displaystyle U} , then so are f + g {\displaystyle f+g} , f − g {\displaystyle f-g} , f g {\displaystyle fg} , and f ∘ g {\displaystyle f\circ g} . Furthermore, f / g {\displaystyle f/g}
SECTION 10
#1732905187922960-440: A function is continuous, monotropic , and has a derivative, when the variable moves in a certain part of the [complex] plane , we say that it is holomorphic in that part of the plane. We mean by this name that it resembles entire functions which enjoy these properties in the full extent of the plane. ... [A rational fraction admits as poles the roots of the denominator; it is a holomorphic function in all that part of
1040-684: A function with negative (or positive) Schwarzian will remain negative (resp. positive), a fact of use in the study of one-dimensional dynamics . 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, S f + ( f ′ ) 2 ( ( S f − 1 ) ∘ f ) = 0 {\displaystyle Sf+(f')^{2}((Sf^{-1})\circ f)=0} . This follows from
1120-1037: A holomorphic function can be extended to the infinite-dimensional spaces of functional analysis . For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers. Lorsqu'une fonction est continue, monotrope, et a une dérivée, quand la variable se meut dans une certaine partie du plan, nous dirons qu'elle est holomorphe dans cette partie du plan. Nous indiquons par cette dénomination qu'elle est semblable aux fonctions entières qui jouissent de ces propriétés dans toute l'étendue du plan. [...] Une fraction rationnelle admet comme pôles les racines du dénominateur; c'est une fonction holomorphe dans toute partie du plan qui ne contient aucun de ses pôles. Lorsqu'une fonction est holomorphe dans une partie du plan, excepté en certains pôles, nous dirons qu'elle est méromorphe dans cette partie du plan, c'est-à-dire semblable aux fractions rationnelles. [When
1200-468: A holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points a 1 , … , a n {\displaystyle a_{1},\ldots ,a_{n}} on the real axis. Then p ( x ) = S ( f )( x ) is real-valued when x is real and different from all the points a i . By the Schwarz reflection principle p ( x ) extends to
1280-450: A rational function on the complex plane with a double pole at a i : The real numbers β i are called accessory parameters . They are subject to three linear constraints: which correspond to the vanishing of the coefficients of z − 1 , z − 2 {\displaystyle z^{-1},z^{-2}} and z − 3 {\displaystyle z^{-3}} in
1360-801: A real parameter t {\displaystyle t} . Let X {\displaystyle X} denote the two-dimensional space of solutions. For t ∈ R {\displaystyle t\in \mathbb {R} } , let ev t : X → R {\displaystyle \operatorname {ev} _{t}:X\to \mathbb {R} } be the evaluation functional ev t ( x ) = x ( t ) {\displaystyle \operatorname {ev} _{t}(x)=x(t)} . The map t ↦ ker ( ev t ) {\displaystyle t\mapsto \operatorname {ker} (\operatorname {ev} _{t})} gives, for each point t {\displaystyle t} of
1440-479: A set is called a domain of holomorphy . A complex differential ( p , 0 ) {\displaystyle (p,0)} -form α {\displaystyle \alpha } is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: ∂ ¯ α = 0 {\displaystyle {\bar {\partial }}\alpha =0} . The concept of
1520-462: A single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex Reinhardt domains , the simplest example of which is a polydisk . However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such
1600-460: A suitable choice of q ( z ) , the ordinary differential equation takes the form Thus q ( z ) u i ( z ) {\displaystyle q(z)u_{i}(z)} are eigenfunctions of a Sturm–Liouville equation on the interval [ a i , a i + 1 ] {\displaystyle [a_{i},a_{i+1}]} . By the Sturm separation theorem ,
1680-457: A translation, rotation, and scaling of the complex plane, ( M − 1 ∘ f ) ( z ) = {\displaystyle (M^{-1}\circ f)(z)={}} z + z 3 + O ( z 4 ) {\displaystyle z+z^{3}+O(z^{4})} in a neighborhood of zero. Up to third order this function maps the circle of radius r {\displaystyle r} to
SECTION 20
#17329051879221760-405: A triangle, when n = 3 , there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and f ( z ) 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 λ . Writing U ( z ) = q ( z ) u ( z ) for
1840-451: Is analytic at a point p {\displaystyle p} if there exists a neighbourhood of p {\displaystyle p} in which f {\displaystyle f} is equal to a convergent power series in n {\displaystyle n} complex variables; the function f {\displaystyle f}
1920-574: Is antiholomorphic .) The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function f : ( z 1 , z 2 , … , z n ) ↦ f ( z 1 , z 2 , … , z n ) {\displaystyle f\colon (z_{1},z_{2},\ldots ,z_{n})\mapsto f(z_{1},z_{2},\ldots ,z_{n})} in n {\displaystyle n} complex variables
2000-430: Is complex differentiable at every point of U {\displaystyle U} . A function f {\displaystyle f} is holomorphic at a point z 0 {\displaystyle z_{0}} if it is holomorphic on some neighbourhood of z 0 {\displaystyle z_{0}} . A function
2080-516: Is complex differentiable at 0 {\displaystyle 0} , but is not complex differentiable anywhere else, esp. including in no place close to 0 {\displaystyle 0} (see the Cauchy–Riemann equations, below). So, it is not holomorphic at 0 {\displaystyle 0} . The relationship between real differentiability and complex differentiability
2160-464: Is holomorphic in an open subset U {\displaystyle U} of C n {\displaystyle \mathbb {C} ^{n}} if it is analytic at each point in U {\displaystyle U} . Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f {\displaystyle f} , this
2240-525: Is holomorphic on some non-open set A {\displaystyle A} if it is holomorphic at every point of A {\displaystyle A} . A function may be complex differentiable at a point but not holomorphic at this point. For example, the function f ( z ) = | z | l 2 = z z ¯ {\displaystyle \textstyle f(z)=|z|{\vphantom {l}}^{2}=z{\bar {z}}}
2320-468: Is meromorphic on C {\displaystyle \mathbb {C} } .) As a consequence of the Cauchy–Riemann equations , any real-valued holomorphic function must be constant . Therefore, the absolute value | z | {\displaystyle |z|} , the argument arg z {\displaystyle \arg z} ,
2400-417: Is a harmonic function on R 2 {\displaystyle \textstyle \mathbb {R} ^{2}} (each satisfies Laplace's equation ∇ 2 u = ∇ 2 v = 0 {\displaystyle \textstyle \nabla ^{2}u=\nabla ^{2}v=0} ), with v {\displaystyle v}
2480-468: Is a complex domain, f : U → C {\displaystyle f\colon U\to \mathbb {C} } is a holomorphic function and the closed disk D ≡ { z : {\displaystyle D\equiv \{z:} is completely contained in U {\displaystyle U} . Let γ {\displaystyle \gamma } be
Schwarzian derivative - Misplaced Pages Continue
2560-570: Is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use. Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero. That is, if functions f {\displaystyle f} and g {\displaystyle g} are holomorphic in
2640-470: 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
2720-493: Is equal to f ′ ( z ) d z {\displaystyle f'(z)\,\mathrm {d} z} for some continuous function f ′ {\displaystyle f'} . It follows from that d f ′ {\displaystyle \mathrm {d} f'} is also proportional to d z {\displaystyle \mathrm {d} z} , implying that
2800-456: Is equal to its end point, and f : U → C {\displaystyle f\colon U\to \mathbb {C} } is a holomorphic function. Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary. Furthermore: Suppose U ⊂ C {\displaystyle U\subset \mathbb {C} }
2880-440: Is equivalent to f {\displaystyle f} being holomorphic in each variable separately (meaning that if any n − 1 {\displaystyle n-1} coordinates are fixed, then the restriction of f {\displaystyle f} is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that
2960-498: Is functionally independent from z ¯ {\displaystyle {\bar {z}}} , the complex conjugate of z {\displaystyle z} . If continuity is not given, the converse is not necessarily true. A simple converse is that if u {\displaystyle u} and v {\displaystyle v} have continuous first partial derivatives and satisfy
3040-424: Is holomorphic if g {\displaystyle g} has no zeros in U {\displaystyle U} ; otherwise it is meromorphic . If one identifies C {\displaystyle \mathbb {C} } with the real plane R 2 {\displaystyle \textstyle \mathbb {R} ^{2}} , then
3120-765: Is holomorphic on the simply connected region U {\displaystyle U} is also integrable on U {\displaystyle U} . (For a path γ {\displaystyle \gamma } from z 0 {\displaystyle z_{0}} to z {\displaystyle z} lying entirely in U {\displaystyle U} , define F γ ( z ) = F ( 0 ) + ∫ γ f d z {\displaystyle F_{\gamma }(z)=F(0)+\int _{\gamma }f\,\mathrm {d} z} ; in light of
3200-405: Is holomorphic. Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes: Here γ {\displaystyle \gamma } is a rectifiable path in a simply connected complex domain U ⊂ C {\displaystyle U\subset \mathbb {C} } whose start point
3280-438: Is not zero, holomorphic functions are conformal : they preserve angles and the shape (but not size) of small figures. Every holomorphic function is analytic . That is, a holomorphic function f {\displaystyle f} has derivatives of every order at each point a {\displaystyle a} in its domain, and it coincides with its own Taylor series at
Schwarzian derivative - Misplaced Pages Continue
3360-495: 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 f ~ {\displaystyle {\tilde {f}}} onto itself. In fact f ~ {\displaystyle {\tilde {f}}}
3440-568: Is the following: If a complex function f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)} is holomorphic, then u {\displaystyle u} and v {\displaystyle v} have first partial derivatives with respect to x {\displaystyle x} and y {\displaystyle y} , and satisfy
3520-421: Is the real part of a holomorphic function: If v {\displaystyle v} is the harmonic conjugate of u {\displaystyle u} , unique up to a constant, then f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)}
3600-435: Is the whole complex plane is called an entire function . The phrase "holomorphic at a point z 0 {\displaystyle z_{0}} " means not just differentiable at z 0 {\displaystyle z_{0}} , but differentiable everywhere within some close neighbourhood of z 0 {\displaystyle z_{0}} in
3680-462: Is therefore holomorphic wherever the logarithm log z {\displaystyle \log z} is. The reciprocal function 1 z {\displaystyle {\tfrac {1}{z}}} is holomorphic on C ∖ { 0 } {\displaystyle \mathbb {C} \smallsetminus \{0\}} . (The reciprocal function, and any other rational function ,
3760-466: The A , B , C {\displaystyle A,B,C} that make the first three coefficients of M − 1 ∘ f {\displaystyle M^{-1}\circ f} equal to 0 , 1 , 0. {\displaystyle 0,1,0.} Plugging it into the fourth coefficient, we get a = ( S f ) ( z 0 ) {\displaystyle a=(Sf)(z_{0})} . After
3840-502: The Cauchy–Riemann equations : or, equivalently, the Wirtinger derivative of f {\displaystyle f} with respect to z ¯ {\displaystyle {\bar {z}}} , the complex conjugate of z {\displaystyle z} , is zero: which is to say that, roughly, f {\displaystyle f}
3920-979: The Jordan curve theorem and the generalized Stokes' theorem , F γ ( z ) {\displaystyle F_{\gamma }(z)} is independent of the particular choice of path γ {\displaystyle \gamma } , and thus F ( z ) {\displaystyle F(z)} is a well-defined function on U {\displaystyle U} having d F = f d z {\displaystyle \mathrm {d} F=f\,\mathrm {d} z} or f = d F d z {\displaystyle f={\frac {\mathrm {d} F}{\mathrm {d} z}}} . All polynomial functions in z {\displaystyle z} with complex coefficients are entire functions (holomorphic in
4000-466: 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
4080-634: The complex logarithm function log z {\displaystyle \log z} is holomorphic on the domain C ∖ { z ∈ R : z ≤ 0 } {\displaystyle \mathbb {C} \smallsetminus \{z\in \mathbb {R} :z\leq 0\}} . The square root function can be defined as z ≡ exp ( 1 2 log z ) {\displaystyle {\sqrt {z}}\equiv \exp {\bigl (}{\tfrac {1}{2}}\log z{\bigr )}} and
SECTION 50
#17329051879224160-496: 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 π α 1 , … , π α n {\displaystyle \pi \alpha _{1},\ldots ,\pi \alpha _{n}} in clockwise order. Let f : H → Δ be
4240-399: The harmonic conjugate of u {\displaystyle u} . Conversely, every harmonic function u ( x , y ) {\displaystyle u(x,y)} on a simply connected domain Ω ⊂ R 2 {\displaystyle \textstyle \Omega \subset \mathbb {R} ^{2}}
4320-511: The real part Re ( z ) {\displaystyle \operatorname {Re} (z)} and the imaginary part Im ( z ) {\displaystyle \operatorname {Im} (z)} are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate z ¯ . {\displaystyle {\bar {z}}.} (The complex conjugate
4400-545: The seminorms being the suprema on compact subsets . From a geometric perspective, a function f {\displaystyle f} is holomorphic at z 0 {\displaystyle z_{0}} if and only if its exterior derivative d f {\displaystyle \mathrm {d} f} in a neighbourhood U {\displaystyle U} of z 0 {\displaystyle z_{0}}
4480-578: The Cauchy–Riemann equations, then f {\displaystyle f} is holomorphic. The term holomorphic was introduced in 1875 by Charles Briot and Jean-Claude Bouquet , two of Augustin-Louis Cauchy 's students, and derives from the Greek ὅλος ( hólos ) meaning "whole", and μορφή ( morphḗ ) meaning "form" or "appearance" or "type", in contrast to the term meromorphic derived from μέρος ( méros ) meaning "part". A holomorphic function resembles an entire function ("whole") in
4560-552: The Cauchy–Riemann equations, then f {\displaystyle f} is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem : if f {\displaystyle f} is continuous, u {\displaystyle u} and v {\displaystyle v} have first partial derivatives (but not necessarily continuous), and they satisfy
4640-523: The Schwarzian derivative precisely measures the degree to which a function fails to be a Möbius transformation. If g is a Möbius transformation, then the composition g o f has the same Schwarzian derivative as f ; and on the other hand, the Schwarzian derivative of f o g is given by the chain rule More generally, for any sufficiently differentiable functions f and g When f and g are smooth real-valued functions, this implies that all iterations of
4720-499: The Schwarzian, if two diffeomorphisms of a common open interval into R P 1 {\displaystyle \mathbb {RP} ^{1}} 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 R P 1 {\displaystyle \mathbb {RP} ^{1}} . Alternatively, consider
4800-531: The chain rule above. William Thurston interprets the Schwarzian derivative as a measure of how much a conformal map deviates from a Möbius transformation . Let f {\displaystyle f} be a conformal mapping in a neighborhood of z 0 ∈ C . {\displaystyle z_{0}\in \mathbb {C} .} Then there exists a unique Möbius transformation M {\displaystyle M} such that M , f {\displaystyle M,f} has
4880-457: The circle forming the boundary of D {\displaystyle D} . Then for every a {\displaystyle a} in the interior of D {\displaystyle D} : where the contour integral is taken counter-clockwise . The derivative f ′ ( a ) {\displaystyle {f'}(a)} can be written as
SECTION 60
#17329051879224960-414: The complex plane. Given a complex-valued function f {\displaystyle f} of a single complex variable, the derivative of f {\displaystyle f} at a point z 0 {\displaystyle z_{0}} in its domain is defined as the limit This is the same definition as for the derivative of
5040-507: The continuity assumption is unnecessary: f {\displaystyle f} is holomorphic if and only if it is holomorphic in each variable separately. More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions. Functions of several complex variables are in some basic ways more complicated than functions of
5120-591: The derivative d f ′ {\displaystyle \mathrm {d} f'} is itself holomorphic and thus that f {\displaystyle f} is infinitely differentiable. Similarly, d ( f d z ) = f ′ d z ∧ d z = 0 {\displaystyle \mathrm {d} (f\,\mathrm {d} z)=f'\,\mathrm {d} z\wedge \mathrm {d} z=0} implies that any function f {\displaystyle f} that
5200-447: The domain of X {\displaystyle X} , a one-dimensional linear subspace of X {\displaystyle X} . 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 2 p ( t ) {\displaystyle 2p(t)} ( Ovsienko & Tabachnikov 2005 ). Owing to this interpretation of
5280-619: The domain on which f 1 ( z ) {\displaystyle f_{1}(z)} and f 2 ( z ) {\displaystyle f_{2}(z)} are defined, and f 2 ( z ) ≠ 0. {\displaystyle f_{2}(z)\neq 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 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} can be found, and furthermore, these are unique up to
5360-409: The eccentricity measures the deviation of f {\displaystyle f} from a Möbius transform. Consider the linear second-order ordinary differential equation x ″ ( t ) + p ( t ) x ( t ) = 0 {\displaystyle x''(t)+p(t)x(t)=0} where x {\displaystyle x} is a real-valued function of
5440-488: The expansion of p ( z ) around z = ∞ . The mapping f ( z ) can then be written as where u 1 ( z ) {\displaystyle u_{1}(z)} and u 2 ( z ) {\displaystyle u_{2}(z)} 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
5520-494: The holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations , a set of two partial differential equations . Every holomorphic function can be separated into its real and imaginary parts f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(x+iy)=u(x,y)+i\,v(x,y)} , and each of these
5600-496: The limit exists, f {\displaystyle f} is said to be complex differentiable at z 0 {\displaystyle z_{0}} . This concept of complex differentiability shares several properties with real differentiability : It is linear and obeys the product rule , quotient rule , and chain rule . A function is holomorphic on an open set U {\displaystyle U} if it
5680-547: The mapping In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on Diff( S ) with coefficients in F 2 ( S ) . In fact Holomorphic function In mathematics , a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space C n {\displaystyle \mathbb {C} ^{n}} . The existence of
5760-399: The non-vanishing of u 2 ( z ) {\displaystyle 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 half-plane H , onto itself, with two mappings considered to be equivalent if on the boundary one
5840-2004: The parametric curve defined by ( r cos θ + r 3 cos 3 θ , r sin θ + r 3 sin 3 θ ) , {\displaystyle (r\cos \theta +r^{3}\cos 3\theta ,r\sin \theta +r^{3}\sin 3\theta ),} where θ ∈ [ 0 , 2 π ] . {\displaystyle \theta \in [0,2\pi ].} This curve is, up to fourth order, an ellipse with semiaxes r + r 3 {\displaystyle r+r^{3}} and | r − r 3 | {\displaystyle |r-r^{3}|} : ( r cos θ + r 3 cos 3 θ ) 2 ( r + r 3 ) 2 + ( r sin θ + r 3 sin 3 θ ) 2 ( r − r 3 ) 2 = 1 + 8 r 4 sin 2 ( 2 θ ) + O ( r 6 ) ( 1 − r 4 ) 2 → 1 + 8 r 4 sin 2 ( 2 θ ) + O ( r 6 ) {\displaystyle {\begin{aligned}{\frac {(r\cos \theta +r^{3}\cos 3\theta )^{2}}{(r+r^{3})^{2}}}+{\frac {(r\sin \theta +r^{3}\sin 3\theta )^{2}}{(r-r^{3})^{2}}}&={\frac {1+8r^{4}\sin ^{2}(2\theta )+O(r^{6})}{(1-r^{4})^{2}}}\\[5mu]&\rightarrow 1+8r^{4}\sin ^{2}(2\theta )+O(r^{6})\end{aligned}}} as r → 0. {\displaystyle r\rightarrow 0.} Since Möbius transformations always map circles to circles or lines,
5920-551: The same 0, 1, 2-th order derivatives at z 0 . {\displaystyle z_{0}.} Now ( M − 1 ∘ f ) ( z − z 0 ) = z 0 + ( z − z 0 ) + 1 6 a ( z − z 0 ) 3 + ⋯ . {\displaystyle (M^{-1}\circ f)(z-z_{0})=z_{0}+(z-z_{0})+{\tfrac {1}{6}}a(z-z_{0})^{3}+\cdots .} To explicitly solve for
6000-496: The second-order linear ordinary differential equation in the complex plane Let f 1 ( z ) {\displaystyle f_{1}(z)} and f 2 ( z ) {\displaystyle f_{2}(z)} be two linearly independent holomorphic solutions. Then the ratio g ( z ) = f 1 ( z ) / f 2 ( z ) {\displaystyle g(z)=f_{1}(z)/f_{2}(z)} satisfies over
6080-420: The set of holomorphic functions on an open set is a commutative ring and a complex vector space . Additionally, the set of holomorphic functions in an open set U {\displaystyle U} is an integral domain if and only if the open set U {\displaystyle U} is connected. In fact, it is a locally convex topological vector space , with
6160-457: 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
6240-498: The subspace of the universal Teichmüller space invariant under Γ . The holomorphic functions g have the property that 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 finite-dimensional complex vector space of quadratic differentials on S . The transformation property allows the Schwarzian derivative to be interpreted as
6320-415: The unit disc, D , 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 D satisfying then Nehari proved that f is univalent. In particular a sufficient condition for univalence is The Schwarzian derivative and associated second-order ordinary differential equation can be used to determine
6400-948: The whole complex plane C {\displaystyle \mathbb {C} } ), and so are the exponential function exp z {\displaystyle \exp z} and the trigonometric functions cos z = 1 2 ( exp ( + i z ) + exp ( − i z ) ) {\displaystyle \cos {z}={\tfrac {1}{2}}{\bigl (}\exp(+iz)+\exp(-iz){\bigr )}} and sin z = − 1 2 i ( exp ( + i z ) − exp ( − i z ) ) {\displaystyle \sin {z}=-{\tfrac {1}{2}}i{\bigl (}\exp(+iz)-\exp(-iz){\bigr )}} (cf. Euler's formula ). The principal branch of
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