| Clifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Clifford-valued functions that lie in the kernel of the Cauchy-Riemann operator. In this dissertation, we explore the relationships between the harmonic components of monogenic functions and expand upon the notion of conjugate harmonic functions. We show that properties of the even part of a Clifford-valued function determine properties of the odd part and vice versa. We also explore the theory of functions lying in the kernel of a generalized Laplace operator, the lambda-Laplacian. We explore the properties of these so-called lambda-harmonic functions and give the solution to the Dirichlet problem for the lambda-harmonic functions on annular domains in Rn. |
