Schwarzian derivative

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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.

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[edit] Definition

The Schwarzian derivative of a function of one complex variable f is defined by

(Sf)(z) = \left({f''(z) \over f'(z)}\right)' - {1\over 2}\left({f''(z)\over f'(z)}\right)^2
={f'''(z) \over f'(z)}-{3\over 2}\left({f''(z)\over f'(z)}\right)^2

The alternate notation

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

is frequently used.

[edit] Properties

The Schwarzian derivative of a linear fractional transformation g(z) = \frac{az + b}{cz + d} is zero. This suggests the following intuitive interpretation of this operator: Just as the ordinary derivative tells us how well a function is approximated by a linear function, the Schwarzian derivative tells us how well a function is approximated by a fractional linear function.

If g\ is a fractional linear transformation, then the composition g\circ f has the same Schwarzian derivative as f\ . On the other hand, the Schwarzian derivative of f\circ g is given by the remarkable 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 one-dimensional dynamics since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.

The Schwarzian derivative can also be defined as the following limit

(Sf)(y)=6\lim_{x\rightarrow y} \left({f^\prime(x)f^\prime(y)\over(f(x)-f(y))^2}-{1\over(x-y)^2}\right).

[edit] 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

\frac{d^2f}{dz^2}+ Q(z) f(z)=0.

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 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 f1 and f2 can be found, and furthermore, these are unique up to 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.

[edit] Schwarzian derivatives as cocycles

For a one-dimensional manifold M, let Fλ(M) be the space of tensor densities of degree λ on M. The group of diffeomorphisms of M, Diff(M), acts on Fλ(M) via pushforwards. If f is an element of Diff(M) then consider the mapping

f \rightarrow S(f^{-1})\,.

In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on \text{Diff}(\mathbb{RP}^1) with coefficients in F_2(\mathbb{RP}^1). In fact

H^1(\text{Diff}(\mathbb{RP}^1);F_2) = \mathbb{R}

and the 1-cocycle generating the cohomology is f → S(f−1).

There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra Vect(\mathbb{RP}^1) of vector fields. This in turn gives the unique non-trivial central extension of Vect(S1), the Virasoro algebra.

[edit] Inversion formula

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'.

[edit] References

  • V. Ovsienko, S. Tabachnikov : Projective Differential Geometry Old and New, Cambridge University Press, 2005. ISBN 0-521-83186-5 .
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