This book provides a new mathematical theory for the treatment of an ample series of spatial problems of electrodynamics, particle physics, quantum mechanics and elasticity theory. This technique proves to be as powerful for solving the spatial problems of mathematical physics as complex analysis is for solving planar problems.

The main analytic tool of the book, a non-harmonic version of hypercomplex analysis recently developed by the authors, is presented in detail. There are given applications of this theory to the boundary value problems of electrodynamics and elasticity theory as well as to the problem of quark confinement. A new approach to the linearization of special classes of the self-duality equation is also considered. Detailed proofs are given throughout. The book contains an extensive bibliography on closely related topics.

This book will be of particular interest to academic and professional specialists and students in mathematics and physics who are interested in integral representations for partial differential equations. The book is self-contained and could be used as a main reference for special course seminars on the subject.

0 Introduction and some remarks on generalizations of complex analysis -- 1 a-holomorphic function theory -- 1 Algebras of real and complex quaternions -- 2 Helmholtz operator with complex and quaternionic wave number -- 3 Notion of a-holomorphic function -- 4 Integral formulas for a -holomorphic functions -- 5 Boundary value properties of a-holomorphic functions -- 6 Boundary value problems for a -holomorphic functions -- 7 Vectorial reformulation of some quaternionic objects and results -- 2 Electrodynamical models -- 8 The classical Maxwell equations -- 9 Relation between electromagnetic fields and a -holomorphic functions -- 10 Integral representations of electrodynamical quantities -- 11 Boundary value problems for time-harmonic electromagnetic fields -- 3 Massive spinor fields -- 12 The classical Dirac equation and its biquaternionic reformulation -- 13 Integral representations and boundary value problems for time-harmonic spinor fields -- 14 Boundary integral criteria for the MIT bag model -- 4 Hypercomplex factorization, systems of non-linear partial differentis equations generated by Futer-type operators -- 15 Notion of hypercomplex factorization and applications to mathematical physics -- 16 Self-duality equation and other systems of non-linear partial differential equations generated by the Futer-type operators -- Appendices -- 1 Real quaternions -- 2 Representations of real quaternions -- 3 Complex quaternions -- 4 a -holomorphic functions of two real variables -- Bibliography -- Index.