In mathematics, the Schwarzian derivative 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 series. Mathematics is the study of quantity, structure, space and change. ... Möbius transformations should not be confused with the Möbius transform. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ...

Definition

The Schwarzian derivative of a function of one complex variable f is defined by Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...

The alternate notation

{f,z} = (Sf)(z)

is frequently used.

Properties

The Schwarzian derivative of a linear fractional transformation Möbius transformations should not be confused with the Möbius transform. ...

g(z) = (az + b) / (cz + d)

is zero.

If we follow a function f by a fractional linear transformation g then the composition has the same Schwarzian derivative as f.

On the other hand the Schwarzian derivative of , where g is again fractional linear, is given by the remarkable chain-like rule

More generally for any sufficiently differentiable f and g

Just as the ordinary derivative tells us how a function can be approximated by a linear function, the Schwarzian derivative tells us how a function can be approximated by a fractional linear function.

The Schwarzian derivative can also be defined as the following limit

Differential equation

The Schwarzian derivative has a curious interplay with second-order linear ordinary differential equations. Let f₁(z) and f₂(z) be two linearly independent holomorphic solutions of In mathematics, and in particular analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. ... In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Then the ratio g(z) = f₁(z) / f₂(z) satisfies

(Sg)(z) = 2Q(z)

over the domain on which f₁(z) and f₂(z) 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 f₁ and f₂ can be found, and furthermore, these are unique up to a common scale factor. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

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. In mathematics, the hypergeometric differential equation is a second-order linear ordinary differential equation (ODE) whose solutions are given by the hypergeometric series. ...

Schwarzian derivatives as cocycles

For a one-dimensional complex manifold M, let F_λ(M) be the space of tensor densities of degree λ on M. The group of diffeomorphisms of M, Diff(M), acts on F_λ(M) via pushforwards. If f is an element of Diff(M) then consider the mapping In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...

f − > S(f ^{− 1}).

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 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. ...

and the 1-cocycle generating the cohomology is f − > S(f ^{− 1}).

There is an infinitesimal version of this result giving a 1-cocyle for the Lie algebra Vect() of vector fields. This in turns gives the unique non-trivial central extension of Vect(S¹), the Virasoro algebra. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, the Virasoro group is a central extension of the orientation-preserving diffeomorphism group of the circle. ...

Inversion 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 v'(w) = 1 / w'. In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

References

• V. Ovsienko, S. Tabachnikov : Projective Differential Geometry Old and New, Cambridge University Press, 2005. ISBN 0521831865 .