This book covers recent developments in pseudoanalytic function theory including applications and multidimensional generalizations. Written at an undergraduate level, it is the first book of the theory applied to many models of mathematical physics.

Pseudoanalytic function theory generalizes and preserves many crucial features of complex analytic function theory. The Cauchy-Riemann system is replaced by a much more general first-order system with variable coefficients which turns out to be closely related to important equations of mathematical physics. This relation supplies powerful tools for studying and solving Schrödinger, Dirac, Maxwell, Klein-Gordon and other equations with the aid of complex-analytic methods.

The book is dedicated to these recent developments in pseudoanalytic function theory and their applications as well as to multidimensional generalizations.

It is directed to undergraduates, graduate students and researchers interested in complex-analytic methods, solution techniques for equations of mathematical physics, partial and ordinary differential equations.

First presentation in book of pseudoanalytic function theory applied to a variety of models of mathematical physics Presents new methods for studying models of mathematical physics Clearly written, at the level of an undergraduate student acquainted with complex analysis and basics of differential equations Includes supplementary material: sn.pub/extras

Autorentext

Alexey N. Karapetyants is professor of the Institute of Mathematics, Mechanics and Computer Sciences and director of Regional Mathematical Center at the Southern Federal University, Rostov-on-Don, Russia. His research interests include operator theory, mathematical physics, harmonic analysis: real and complex variable methods, and functional analysis. Vladislav V. Kravchenko is a researcher in the Department of Mathematics of Center for Research and Advanced Studies of the National Polytechnic Institute, Campus Queretaro, Mexico. His research interests include mathematical physics, differential equations, complex and hypercomplex analysis, spectral theory, and inverse problems.

Inhalt
Pseudoanalytic Function Theory and Second-order Elliptic Equations.- Definitions and Results from Bers' Theory.- Solutions of Second-order Elliptic Equations as Real Components of Complex Pseudoanalytic Functions.- Formal Powers.- Cauchy's Integral Formula.- Complex Riccati Equation.- Applications to Sturm-Liouville Theory.- A Representation for Solutions of the Sturm-Liouville Equation.- Spectral Problems and Darboux Transformation.- Applications to Real First-order Systems.- Beltrami Fields.- Static Maxwell System in Axially Symmetric Inhomogeneous Media.- Hyperbolic Pseudoanalytic Functions.- Hyperbolic Numbers and Analytic Functions.- Hyperbolic Pseudoanalytic Functions.- Relationship between Hyperbolic Pseudoanalytic Functions and Solutions of the Klein-Gordon Equation.- Bicomplex and Biquaternionic Pseudoanalytic Functions and Applications.- The Dirac Equation.- Complex Second-order Elliptic Equations and Bicomplex Pseudoanalytic Functions.- Multidimensional Second-order Equations.