Application of Holomorphic Functions in Two and Higher Dimensions

This book§presents applications of hypercomplex analysis to boundary value and§initial-boundary value problems from various areas of mathematical physics.§Given that quaternion and Clifford analysis offer natural and intelligent ways§to enter into higher dimensions, it starts with quaternion and Clifford§versions of complex function theory including series expansions with Appell§polynomials, as well as Taylor and Laurent series. Several necessary function§spaces are introduced, and an operator calculus based on modifications of the§Dirac, Cauchy-Fueter, and Teodorescu operators and different decompositions of§quaternion Hilbert spaces are proved. Finally, hypercomplex Fourier§transforms are studied in detail.§§All this is§then applied to first-order partial differential equations such as the Maxwell§equations, the Carleman-Bers-Vekua system, the Schrödinger equation, and the Beltrami§equation. The higher-order equations start with Riccati-type equations. Further§topics include spatial fluid flow problems, image and multi-channel processing,§image diffusion, linear scale invariant filtering, and others. One of the§highlights is the derivation of the three-dimensional Kolosov-Mushkelishvili§formulas in linear elasticity.§§Throughout the book the authors endeavor to§present historical references and important personalities. The book is intended§for a wide audience in the mathematical and engineering sciences and is§accessible to readers with a basic grasp of real, complex, and functional§analysis.§