Singularities of slice regular functions. (English) Zbl 1253.30076
Summary: Beginning in 2006, G. Gentili and D. C. Struppa [C. R., Math., Acad. Sci. Paris 342, No. 10, 741–744 (2006; Zbl 1105.30037)] developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball \(B(0, R)\) centered at \(0\) the set of regular functions coincides with that of quaternionic power series \(\sum_{n \in \mathbb N}q^n a_n\) converging in \(B(0, R)\). In 2009 the author [Complex Var. Elliptic Equ. 54, No. 11, 1001–1018 (2009; Zbl 1177.30071)] proposed a classification of singularities of regular functions as removable, essential or as poles, and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls \(B(0, R)\). Quite recently, F. Colombo, G. Gentili and I. Sabadini [Ann. Global Anal. Geom. 37, No. 4, 361–378 (2010; Zbl 1193.30069)] and the same authors in collaboration with D. C. Struppa [Adv. Math. 222, No. 5, 1793–1808 (2009; Zbl 1179.30052)] identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent-type expansions at points \(p\) other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in [Stoppato, loc. cit]. Poles are studied, as well as essential singularities, for which a version of the Casorati-Weierstrass Theorem is proven.
MSC:
| 30G35 | Functions of hypercomplex variables and generalized variables |
| 30D30 | Meromorphic functions of one complex variable (general theory) |
| 30B10 | Power series (including lacunary series) in one complex variable |
References:
| [1] | Cohn, Skew Fields: Theory of General Division Rings, Encyclopedia of Mathematics and its Applications (1995) · doi:10.1017/CBO9781139087193 |
| [2] | Colombo, A Cauchy kernel for slice regular functions, Ann. Global Anal. Geom 37 (4) pp 361– (2010) · Zbl 1193.30069 · doi:10.1007/s10455-009-9191-7 |
| [3] | Colombo, Extension results for slice regular functions of a quaternionic variable, Adv. Math 222 pp 1793– (2009) · Zbl 1179.30052 · doi:10.1016/j.aim.2009.06.015 |
| [4] | Colombo, Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics (2004) · Zbl 1064.30049 · doi:10.1007/978-0-8176-8166-1 |
| [5] | Cullen, An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J 32 pp 139– (1965) · Zbl 0173.09001 · doi:10.1215/S0012-7094-65-03212-6 |
| [6] | Fueter, Die Funktionentheorie der Differentialgleichungen {\(\Delta\)}u = 0 und {\(\Delta\)}{\(\Delta\)}u = 0 mit vier reellen Variablen, Comment. Math. Helv 7 (1) pp 307– (1934) · Zbl 0012.01704 · doi:10.1007/BF01292723 |
| [7] | Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv 8 (1) pp 371– (1935) · JFM 62.0120.04 · doi:10.1007/BF01199562 |
| [8] | Gentili, Zeros of regular functions and polynomials of a quaternionic variable, Mich. Math. J 56 (3) pp 655– (2008) · Zbl 1184.30048 · doi:10.1307/mmj/1231770366 |
| [9] | Gentili, The open mapping theorem for regular quaternionic functions, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5.) VIII (4) pp 805– (2009) · Zbl 1201.30067 |
| [10] | Gentili, Trends in Mathematics pp 95– (2011) |
| [11] | Gentili, Power series and analyticity over the quaternions, Math. Ann 352 (1) pp 113– (2012) · Zbl 1262.30053 · doi:10.1007/s00208-010-0631-2 |
| [12] | Gentili, Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Pari 342 (10) pp 741– (2006) · Zbl 1105.30037 · doi:10.1016/j.crma.2006.03.015 |
| [13] | Gentili, A new theory of regular functions of a quaternionic variable, Adv. Math 216 (1) pp 279– (2007) · Zbl 1124.30015 · doi:10.1016/j.aim.2007.05.010 |
| [14] | Gentili, On the multiplicity of zeroes of polynomials with quaternionic coefficients, Milan J. Math 76 pp 15– (2008) · Zbl 1194.30054 · doi:10.1007/s00032-008-0093-0 |
| [15] | Kravchenko, Integral Representations for Spatial Models of Mathematical Physics, Pitman Research Notes in Mathematics Series (1996) · Zbl 0872.35001 |
| [16] | Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics (1999) · Zbl 0911.16001 · doi:10.1007/978-1-4612-0525-8 |
| [17] | Pogorui, On the structure of the set of zeros of quaternionic polynomials, Complex Variables, Theory Appl 49 (6) pp 379– (2004) · Zbl 1160.30353 · doi:10.1080/0278107042000220276 |
| [18] | Rowen, Ring Theory, student edition (1991) |
| [19] | Stoppato, Poles of regular quaternionic functions, Complex Var. Elliptic Equ 54 (11) pp 1001– (2009) · Zbl 1177.30071 · doi:10.1080/17476930903275938 |
| [20] | Stoppato, Regular Moebius transformations of the space of quaternions, Ann. Global Anal. Geom 39 (4) pp 387– (2011) · Zbl 1214.30044 · doi:10.1007/s10455-010-9238-9 |
| [21] | Sudbery, Quaternionic analysis, Math. Proc. Camb. Philos. Soc 85 (2) pp 199– (1979) · Zbl 0399.30038 · doi:10.1017/S0305004100055638 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
