Applied Pseudoanalytic Function Theory

Cover -- Contents -- Foreword -- Part I: Pseudoanalytic Function Theory and Second-order Elliptic Equations -- 1 Introduction -- 2 Definitions and Results from Bers Theory -- 2.1 Generating pairs and differentiation -- 2.2 Pseudoanalytic functions -- 2.3 Derivatives and integrals of pseudoanalytic functions -- 3 Solutions of Second-order Elliptic Equations as Real Components of Complex Pseudoanalytic Functions -- 3.1 Factorization of the stationary Schr246;dinger operator -- 3.2 Factorization of the operator div p grad+q -- 3.3 Conjugate metaharmonic functions -- 3.4 The main Vekua equation -- 3.5 Cauchys integral theorem for the Schr246;dinger equation -- 3.6 p-analytic functions -- 4 Formal Powers -- 4.1 Definition -- 4.2 An important special case -- 4.3 Similarity principle -- 4.4 Taylor series in formal powers -- 4.5 The Runge theorem -- 4.6 Complete systems of solutions for second-order equations -- 4.7 A remark on orthogonal coordinate systems in a plane -- 4.8 Explicit construction of a generating sequence -- 4.9 Explicit construction of complete systems of solutions of second-order elliptic equations -- 4.10 Numerical solution of boundary value problems -- 5 Cauchys Integral Formula -- 5.1 Preliminary information on the Cauchy integral formula for pseudoanalytic functions -- 5.2 Relation between the main Vekua equation and the system describing p-analytic functions -- 5.3 The transplant operator -- 5.4 Cauchy integral formulas for xk-analytic functions -- 6 Complex Riccati Equation -- 6.1 Preliminary notes -- 6.2 The two-dimensional stationary Schr246;dinger equation and the complex Riccati equation -- 6.3 Generalizations of classical theorems -- 6.4 Cauchys integral theorem -- Part II: Applications to Sturm-Liouville Theory -- 7 A Representation for Solutions of the Sturm-Liouville Equation -- 7.1 Solving the one-dimensional Schr246;dinger equation -- 7.2 The 8220;+8221;-case -- 7.3 Two sets of Taylor coeffcients -- 7.4 Solution of the one-dimensional Schr246;dinger equation -- 7.5 Validating the result -- 7.6 The 8220;-8221; case -- 7.7 Complex potential -- 7.8 Solution of the Sturm-Liouville equation -- 7.9 Numerical method for solving Sturm-Liouville equations -- 8 Spectral Problems and Darboux Transformation -- 8.1 Sturm-Liouville problem as a problem of finding zeros of an analytic function -- 8.2 Numerical method for solving Sturm-Liouville problems -- 8.3 A remark on the Darboux transformation -- Part III: Applications to Real First-order Systems -- 9 Beltrami Fields -- 9.1 Description of the result -- 9.2 Reduction of (9.1) to a Vekua equation -- 9.3 Solution in the case when 945; is a function of one Cartesian variable -- 10 Static Maxwell System in Axially Symmetric Inhomogeneous Media -- 10.1 Meridional and transverse fields -- 10.2 Reduction of the static Maxwell system to p-analytic functions -- 10.3 Construction of formal powers -- Part IV: Hyperbolic Pseudoanalytic Functions -- 11 Hyperbolic Numbers and Analytic Functions -- 12 Hyperbolic Pseudoanalytic Functions -- 12.1 Differential operators -- 12.2 Hyperbolic pseudoanalytic function theory -- 12.3 Generating sequences -- 13 Relationship between Hyperbolic Pseudoanalytic Functions and Solutions of the Klein-Gordon Equation -- 13.1 Factorization of the Klein-Gordon equation -- 13.2 The main hyperbolic Vekua equation -- 13.3 Generating sequence