Non-(k)-monogenic points of functions of a quaternion variable

Abstract

The concept of (k)-monogeneity for functions with values in the algebra ℜ of real quaternions has been introduced in [1]. Its definition and some results concerning Cauchy's Formula, the Taylor expansion and Weierstrass's convergence theorem are recalled in a first paragraph; for the proofs we refer the reader to [1] and [2].

We then continue the study of such functions by examining their behaviour in the neighbourhood of singular points. This gives rise to Laurent's expansion (§ 3), the notions of (k)-pole, essential-non-(k)-monogenic point and removable singularity (§§ 4 and 6), Mittag-Leffler's theorem on (k)-meromorphic functions (§ 5) and a finite residue theory (§ 7).