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Throughout the book the authors endeavor to present historical
references and important personalities.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.
