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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Definition
The Schwarzian derivative of a function of one complex variable <math>f<math> is defined by
- <math>(Sf)(z) =
\left({f''(z) \over f'(z)}\right)' - {1\over 2}\left({f''(z)\over f'(z)}\right)^2<math>
- <math>={f'''(z) \over f'(z)}-{3\over 2}\left({f''(z)\over f'(z)}\right)^2<math>
The alternate notation
- <math>\{f,z\} = (Sf)(z)<math>
is frequently used.
Properties
The Schwarzian derivative of a linear fractional transformation
- <math>g(z)=(az+b)/(cz+d)<math>
is zero.
If we follow a function <math>f<math> by a fractional linear transformation <math>g<math> then the composition <math>g\circ f<math> has the same Schwarzian derivative as <math>f<math>.
On the other hand the Schwarzian derivative of <math>f\circ g<math>, where <math>g<math> is again fractional linear, is given by the remarkable chain-like rule
- <math>(S(f\circ g))(z)=g'(z)^2(Sf)(g(z)).<math>
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
- <math>(Sf)(y)=6\lim_{x\rightarrow y} \left({f'(x)f'(y)\over(f(x)-f(y))^2}-{1\over(x-y)^2}\right).<math>
Differential equation
The Schwarzian derivative has a curious interplay with second-order linear ordinary differential equations. Let <math>f_1(z)<math> and <math>f_2(z)<math> be two linearly independent holomorphic solutions of
- <math>\frac{d^2f}{dz^2}+ Q(z) f(z)=0<math>
Then the ratio <math>g(z)=f_1(z)/f_2(z)<math> satisfies
- <math>(Sg)(z) = 2Q(z)<math>
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.
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 sollutions to the hypergeometric equation are related in this way.
Inversion formula
The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
- <math>(Sw)(v) = -\left(\frac{dw}{dv}\right)^2 (Sv)(w)<math>
which follows from the inverse function theorem, namely that <math>v'(w)=1/w'.<math>
References
- V. Ovsienko, S. Tabachnikov : Projective Differential Geometry Old and New, Cambridge University Press, 2005. ISBN 0521831865 .