Applied pseudoanalytic function theory

Contents......Page 8
Foreword......Page 12
1. Introduction......Page 14
Part I: Pseudoanalytic Function Theory and Second-order Elliptic Equations......Page 20
2.1 Generating pairs and differentiation......Page 22
2.2 Pseudoanalytic functions......Page 24
2.3 Derivatives and integrals of pseudoanalytic functions......Page 26
3.1 Factorization of the stationary Schrödinger operator......Page 34
3.2 Factorization of the operator div p grad+q......Page 36
3.3 Conjugate metaharmonic functions......Page 40
3.4 The main Vekua equation......Page 42
3.5 Cauchy\'s integral theorem for the Schrödinger equation......Page 44
3.6 p-analytic functions......Page 45
4.1 Definition......Page 48
4.2 An important special case......Page 50
4.3 Similarity principle......Page 51
4.4 Taylor series in formal powers......Page 54
4.6 Complete systems of solutions for second-order equations......Page 56
4.7 A remark on orthogonal coordinate systems in a plane......Page 58
4.8 Explicit construction of a generating sequence......Page 59
4.9 Explicit construction of complete systems of solutions of second-order elliptic equations......Page 63
4.10 Numerical solution of boundary value problems......Page 65
5.1 Preliminary information on the Cauchy integral formula for pseudoanalytic functions......Page 68
5.2 Relation between the main Vekua equation and the system describing p-analytic functions......Page 70
5.3 The transplant operator......Page 71
5.4 Cauchy integral formulas for x[sup(k)]-analytic functions......Page 73
6.1 Preliminary notes......Page 78
6.2 The two-dimensional stationary Schrödinger equation and the complex Riccati equation......Page 80
6.3 Generalizations of classical theorems......Page 82
6.4 Cauchy\'s integral theorem......Page 84
Part II: Applications to Sturm-Liouville Theory......Page 86
7.1 Solving the one-dimensional Schrödinger equation......Page 88
7.2 The \"+\"-case......Page 89
7.3 Two sets of Taylor coefficients......Page 93
7.4 Solution of the one-dimensional Schrödinger equation......Page 94
7.5 Validating the result......Page 96
7.6 The \"–\" case......Page 97
7.7 Complex potential......Page 98
7.8 Solution of the Sturm-Liouville equation......Page 99
7.9 Numerical method for solving Sturm-Liouville equations......Page 103
8.1 Sturm-Liouville problem as a problem of finding zeros of an analytic function......Page 106
8.2 Numerical method for solving Sturm-Liouville problems......Page 109
8.3 A remark on the Darboux transformation......Page 111
Part III: Applications to Real First-order Systems......Page 114
9.1 Description of the result......Page 116
9.2 Reduction of (9.1) to a Vekua equation......Page 117
9.3 Solution in the case when Α is a function of one Cartesian variable......Page 118
10.1 Meridional and transverse fields......Page 124
10.2 Reduction of the static Maxwell system to p-analytic functions......Page 125
10.3 Construction of formal powers......Page 126
Part IV: Hyperbolic Pseudoanalytic Functions......Page 132
11. Hyperbolic Numbers and Analytic Functions......Page 134
12.1 Differential operators......Page 138
12.2 Hyperbolic pseudoanalytic function theory......Page 139
12.3 Generating sequences......Page 143
13.1 Factorization of the Klein-Gordon equation......Page 146
13.2 The main hyperbolic Vekua equation......Page 148
13.3 Generating sequence for the main hyperbolic Vekua equation......Page 151
Part V: Bicomplex and Biquaternionic Pseudoanalytic Functions and Applications......Page 160
14. The Dirac Equation......Page 162
14.1 Notation......Page 163
14.2 Quaternionic form of the Dirac equation......Page 164
14.3 The Dirac equation in a two-dimensional case as a bicomplex Vekua equation......Page 166
14.4 Some definitions and results from Bers\' theory for bicomplex pseudoanalytic functions......Page 168
14.5 The main bicomplex Vekua equation......Page 171
14.6 Dirac equation with a scalar potential......Page 172
15. Complex Second-order Elliptic Equations and Bicomplex Pseudoanalytic Functions......Page 176
16.1 Factorization......Page 180
16.2 The main quaternionic Vekua equation......Page 182
Open Problems......Page 184
Bibliography......Page 186
R......Page 196
Y......Page 197