3 Introductory Clifford Analysis

We want in this chapter to regard Clifford algebras as natural generalizations of the complex number system. First let us note that if z is a complex number then zz = ‖z‖. Now for a quaternion q we also have qq = ‖q‖. In this way quaternions may be regarded as a generalization of the complex number system. It seems natural to ask if one can extend basic results of one complex variable analysis on holomorphic function theory to four dimensions using quaternions. The answer is yes. This was developed by the Swiss mathematician Rudolph Fueter in the 1930’s and 1940’s and also my Moisil and Teodorecu [29]. See for instance [12]. An excellent review of this work is given in the survey article ”Quaternionic analysis” by A. Sudbery, see [47]. There is also earlier work of Dixon [11]. However in previous lectures we have seen that for a vector x ∈ R when we consider R as embedded in the Clifford algebra Cln then x 2 = −‖x‖. So it seems reasonable to ask if all that is known in the quaternionic setting extends to the Clifford algebra setting. Again the answer is yes. The earlier aspects of this study was developed by amongst others Richard Delanghe [9], Viorel Iftimie [16] and David Hestenes [15]. The subject that has grown from these works is now called Clifford analysis. In more recent times Clifford analysis has found a wealth of unexpected applications in a number of branches of mathematical analysis particularly classical harmonic analysis, see for instance the work of Alan McIntosh and his collaborators, for instance [21, 22], Marius Mitrea [27, 28] and papers in [37]. Links to representation theory and several complex variables may be found in [14, 34, 35, 36] and elsewhere. The purpose of this paper is to present a review of many of the basic aspects of Clifford analysis. Alternative accounts of much of this work together with other related results can be found in [5, 10, 13, 14, 20, 31, 37, 38].