A relation for a complex analytic function w = w(z) that is given by
It was formally defined by H. A. Schwarz in 1869, but had been used before by Kummer in 1834. The Schwarzian derivative plays with respect to the conformal group the same role as the ordinary derivative with respect to translations. It provides a measure on how much a complex function w(z) differs from a Möbius transformation . Lavie's theorem states that any third-order differential operator invariant under the group of projective conformal (or Möbius) transformations is a rational function of the Schwarzian derivative . If w = w p (z) is a Möbius transformation , {w, z} = 0 and
where z → w → s is a sequence of conformal transformations.
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Bibliography
E. Hille, Ordinary Differential Equations in the Complex Domain, New York 1976.
C. Itzykson and J.-M. Drouffe, Statistical Field Theory 2, Cambridge 1989.
H. A. Schwarz, J. Reine Angew. Math. 70 (1869) 105.
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Lozano, Y. et al. (2004). Schwarzian Derivative. In: Duplij, S., Siegel, W., Bagger, J. (eds) Concise Encyclopedia of Supersymmetry. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4522-0_476
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DOI: https://doi.org/10.1007/1-4020-4522-0_476
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