Slice Hyperholomorphic Functions with Values in Some Real Algebras

The main purpose of this chapter is to offer an overview to show how the theory of holomorphic functions of one complex variable can be successfully extended to the setting of real alternative algebras. Thus, the purpose of this chapter is to show how notions of holomorphicity (which will be called hyperholomorphicity in this case) can be properly defined when the field of complex numbers is replaced by what are generically referred to as its hypercomplex generalizations. However, not all possible hypercomplex generalizations will be considered, and, more specifically, this chapter presents various definitions of slice hyperholomorphic functions for functions whose values lie in real alternative algebras with a unit. After recalling the basic definitions from the theory of real alternative algebras, it will be shown how some of the most important hypercomplex algebras fall within this context; in particular the category of real alternative algebras includes quaternions, octonions, and of course Clifford Algebras. On the basis of this theory, this chapter then introduces the basic results in the theory of slice regular functions for quaternions and octonions, as developed in Gentili et al. (Regular Functions of a Quaternionic Variable. Springer, New York, 2013), as well as the corresponding results for slice monogenic functions, as developed in Colombo et al. (Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions. Birkhauser, Basel, 2011). The most important result that will be discussed is the Cauchy formula for slice hyperholomorphic functions, which rests on the so-called noncommutative Cauchy kernel series introduced originally in Colombo and Sabadini (Hypercomplex Analysis. Birkhauser, Basel, 2009). The results that will be described are the foundation for any advanced study of the subject. This chapter then focuses on a different, comprehensive approach to slice hyperholomorphicity, developed in the last few years in Ghiloni and Perotti (Adv Math 226:1662–1691, 2011), and the connections between the two theories are explored and discussed in detail.