Topics in Clifford Analysis

Quaternionic and Clifford analysis are an extension of complex analysis into higher dimensions. The unique starting point of Wolfgang Sprößig’s work was the application of quaternionic analysis to elliptic differential equations and boundary value problems. Over the years, Clifford analysis has become a broad-based theory with a variety of applications both inside and outside of mathematics, such as higher-dimensional function theory, algebraic structures, generalized polynomials, applications of elliptic boundary value problems, wavelets, image processing, numerical and discrete analysis.The aim of this volume is to provide an essential overview of modern topics in Clifford analysis, presented by specialists in the field, and to honor the valued contributions to Clifford analysis made by Wolfgang Sprößig throughout his career.

Part I - Clifford analysis theories.- Cauchy’s formula in Clifford analysis: An Overview.- Quaternionic Hyperbolic Function Theory.- Slice Regular and Harmonicity in Clifford Analysis.- Some notions of Subharmonicity over the Quaternions.- Part II - Applications to Elliptic Partial Differential Equations.- A Fatou Theorem and Poisson’s Integral Representation Formula for Elliptic Systems in the Upper-Half Space.- Hardy Spaces for Three-Dimensional Vekua Equation.- Radial and Angular Derivatives of Distributions.- Applications of Parabolic Dirac Operators to the Instationary Viscous MHD Equations on Conformally Flat Manifolds.- Generalized Riesz Transforms, Quasi-Monogenic Functions and Frames.- Part III - Monogenic Polynomials and Numerical Methods.- Quaternionic Operator Calculus for Boundary Value Problems of Micropolar Elasticity.- Constructive Orthonormalization of Monogenic Polynomials on a Finite Cylinder.- Comments on an Orthogonal Family of Monogenic Functions on Spheroidal Domains.- Newton’s Approach to General Algebra Equations over Clifford Algebras.- Part IV - Differential Geometry.- Connections in Euclidean and Non-Commutative Geometry.- Conformal Parametrisation of Loxodroms by Tripels of Circles.- Automorphic Forms and Dirac Operators on Conformally Flat manifolds.- Higher order Fermionic and Bosonic Operators.- Clifford Möbius Geometry.- Separation of Variables in the Semistable Range.- Variety of Idempotents in Nonassociative Algebras.- Part V - Discrete Clifford Analysis.- Relativistic Wave Equations on the Lattice: An Operational Perspective.- Cauchy-Pompeiu Formula for Discrete Monogenic Functions.- A Useful Transformation for Solving the Discrete Beltrami Equation and Reducing a Difference Equation of Second Order to a System of Equations of First Order.