Generalized Analytic Functions on Riemann Surfaces
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1288 Yuri L. Rodin
Generalized Analytic Functions on Riemann Surfaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Yuri L. Rodin Academy of Sciences of the USSR, Institute of Solid State Physics, Chernogolovka, Moscow Distr. 142432, USSR
Mathematics Subject Classification (1980): 30F30, 30G20
ISBN 3·540·18572·0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18572-0 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PRE F ACE This book presents results arising from several areas of the theory of functions and mathematical physics. The first of these sources, the theory of generalized analytic (pseudo-analytic) functions of L. Bers [a,b] and I.N. Vekua [a,b] has been constructed within the framework of a general interest in different generalizations of analyticity.
It was established that such
fundamental properties of analytic functions as the argument principle, the Liouville theorem and so on are inherent in solutions of all linear elliptic systems of first order with two unknown functions on the plane.
By quasiconformal mappings these systems can be reduced to
the complex Carleman-Bers-Vekua equation au
+
au + bu
=a
(1)
Later the theory of matrix equations (1) was built (W. Wendland [a]). These equations are extremely important for applications (see §12). At the same time on Riemann surfaces the Riemann boundary problem +
-
F (p) = G(p) F (p)
(2)
was studied (A. Grothendieck [a], W. Koppelman [b,c], Yu.L. Rodin [a,c,p], H. Rohrl [a,b] and other authors).
Main facts of the alge-
braic function theory were related with the theory of singular integral operators and the classification problem of vector bundles over Riemann surfaces.
Afterwards this theory found fundamental
physical applications (the Riemann problem method of V.E. Zakharov A.B. Shabat) in the inverse scattering problem, the integrable systems theory and the solitons theory.
At last, recently generalized
analytic functions were used in these areas too (see M.J. Ablowitz, D. Bar Yaacov, A.S. Fokas [a], A.S. Fokas, M.J. Ablowitz [a,b],I.M. Krichever, S.P. Novikov raj, A.V.
[a,b], V.E. Zakharov, S.V.
Manakov [a], V.E. Zakharov, A.V. Mikhailov [a]). These circumstance stimulated the study of generalized analytic functions on Riemann surfaces.
The work was begun by L. Bers [c]
and was continued by W. Koppel
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