Abstract
Our description of the regular transformations of the quaternionic space begins by considering those affine transformations of \(\mathbb{H}\) which are regular.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L.V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1979)
C. Bisi, G. Gentili, Möbius transformations and the Poincaré distance in the quaternionic setting. Indiana Univ. Math. J. 58(6), 2729–2764 (2009)
C. Bisi, G. Gentili, On the geometry of the quaternionic unit disc, in Hypercomplex Analysis and Applications, ed. by I. Sabadini, F. Sommen. Trends in Mathematics (Birkhäuser Basel, 2011), pp. 1–11
C. Bisi, C. Stoppato, Regular vs. classical Möbius transformations of the quaternionic unit ball, in Advances in Hypercomplex Analysis, ed. by G. Gentili, I. Sabadini, M.V. Shapiro, F. Sommen, D.C. Struppa. Springer INdAM Series (Springer, Milan, 2013), pp. 1–13
H. Bohr, A theorem concerning power series. Proc. Lond. Math. Soc. S2-13(1), 1–5 (1914)
D.M. Burns, S.G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7(3), 661–676 (1994)
C. Della Rocchetta, G. Gentili, G. Sarfatti, A Bloch-Landau theorem for slice regular functions, in Advances in Hypercomplex Analysis, ed. by G. Gentili, I. Sabadini, M.V. Shapiro, F. Sommen, D.C. Struppa. Springer INdAM Series (Springer, Milan, 2013), pp. 55–74
C. Della Rocchetta, G. Gentili, G. Sarfatti, The Bohr theorem for slice regular functions. Math. Nachr. 285(17–18), 2093–2105 (2012)
I. Gelfand, V. Retakh, R.L. Wilson, Quaternionic quasideterminants and determinants, in Lie Groups and Symmetric Spaces. American Mathematical Society Translations: Series 2, vol. 210 (American Mathematical Society, Providence, 2003), pp. 111–123
G. Gentili, D.C. Struppa, A new theory of regular functions of a quaternionic variable. Adv. Math. 216(1), 279–301 (2007)
G. Gentili, D.C. Struppa, Lower bounds for polynomials of a quaternionic variable. Proc. Am. Math. Soc. 140(5), 1659–1668 (2012)
G. Gentili, F. Vlacci, Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz Lemma. Indag. Math. (N.S.) 19(4), 535–545 (2008)
G. Gentili, F. Vlacci, On fixed points of regular Möbius transformations over quaternions, in Complex Analysis and Dynamical Systems IV. Part1. Contemporary Mathematics, vol. 553 (American Mathematical Society, Providence, 2011), pp. 75–82
K. Gürlebeck, J. Morais, On the development of Bohr’s phenomenon in the context of quaternionic analysis and related problems, in Algebraic Structures in Partial Differential Equations Related to Complex and Clifford Analysis (Ho Chi Minh City University of Education Press, Ho Chi Minh City, 2010), pp. 9–24
K. Gürlebeck, J. Morais, P. Cerejeiras, Borel-Carathéodory type theorem for monogenic functions. Complex Anal. Oper. Theor. 3(1), 99–112 (2009)
K. Gürlebeck, J.P. Morais, Bohr type theorems for monogenic power series. Comput. Meth. Funct. Theor. 9(2), 633–651 (2009)
A. Herzig, Die Winkelderivierte und das Poisson-Stieltjes-Integral. Math. Z. 46, 129–156 (1940)
S. Migliorini, F. Vlacci, A new rigidity result for holomorphic maps. Indag. Math. (N.S.) 13(4), 537–549 (2002)
W. Rudin, Function theory in the unit ball of { C}n. Grundlehren der Mathematischen Wissenschaften, vol. 241 (Springer, New York, 1980)
C. Stoppato, Regular Moebius transformations of the space of quaternions. Ann. Global Anal. Geom. 39(4) 387–401 (2011)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gentili, G., Stoppato, C., Struppa, D.C. (2013). Fractional Transformations and the Unit Ball. In: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33871-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-33871-7_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33870-0
Online ISBN: 978-3-642-33871-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)