The Pompeiu Formula for Slice Hyperholomorphic Functions
The fundamental result that makes complex analysis into a new discipline, independent from the theory of real variables, is the Cauchy formula, which allows the representation of any holomorphic function through a reproducing holomorphic kernel. This result is in fact an almost immediate application of the Stokes formula, which, in the more general case, offers an integral representation formula for c1 functions. This general representation formula is often known as the Pompeiu formula and can be stated as follows.
Colombo, Fabrizio; Sabadini, Irene; Struppa, Daniel. The Pompeiu formula for slice hyperholomorphic functions. The Michigan Mathematical Journal 60 (2011), no. 1, 163--170. doi:10.1307/mmj/1301586309
University of Michigan, Department of Mathematics