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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.
List of contents
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.
Summary
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.
Additional text
From the reviews:
“This book presents a renaissance of Bers’ and Vekua’s theory. It can be recommended to colleagues which are interested in the plane function theory.” (Michael Reissig, Zentralblatt MATH, Vol. 1182, 2010)
Report
From the reviews:
"This book presents a renaissance of Bers' and Vekua's theory. It can be recommended to colleagues which are interested in the plane function theory." (Michael Reissig, Zentralblatt MATH, Vol. 1182, 2010)
