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Generalized analytic functions on Riemann surfaces. (English) Zbl 0637.30041
Lecture Notes in Mathematics, 1288. Berlin etc.: Springer-Verlag. V, 128 p.; DM 23.00 (1987).
As the author points out in his preface, applications of algebraic function theory and Riemann surfaces to physical problems as well as the appearance of generalized analytic functions in these problems has given rise to studying generalized analytic functions on Riemann surfaces. Thus, in his book the author presents a theory of the Carleman-Bers-Vekua system \[ (*)\quad u_{\bar z}+au+b\bar u=0 \] on Riemann surfaces. He starts with basic facts on the solutions of (*) in the plane. Having generalized (*) so that it becomes a system of a Riemann surface, he concerns himself with integral representation kernels, generalized constants, Riemann boundary value problems on Riemann surfaces and, in particular, establishes a Riemann-Roch theorem. His presentation furthermore deals with systems (*) with singular coefficients and systems (*) on open Riemann surfaces. Finally he indicates some physical applications. Concerning general elliptic systems the author states in the preface that all linear (uniformly) elliptic systems in the plane can be reduced to systems (*) by means of quasiconformal mappings. This is incorrect. In fact, we only know that such a system can be reduced to (*) if certain coefficients have (generalized) partial derivatives; the corresponding transformations, if existing, are in general not quasiconformal. Of course, this misconception does not affect the otherwise very remarkable contents of this book.
Reviewer: H.Renelt

30F30 Differentials on Riemann surfaces
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
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