Slice monogenic functions

By Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa. 2007

204 citations recorded by the source corpus.

In this paper we offer a new definition of monogenicity for functions defined on ℝn+1 with values in the Clifford algebra ℝn following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra ℝn. We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/58c22d9e1510dd506dc767c7384e3a25e09589f2

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In Applied Mathematics

Citations in the corpus

On Some Properties of the Quaternionic Functional Calculus (2009, methodology): “In the paper [11], the theory of quaternionic slice regular functions has been generalized to the Clifford algebras setting.”
Global differential equations for slice regular functions (2014, background): “The related theory of slice monogenic functions on Clifford algebras was introduced by Colombo, Sabadini and Struppa in [2].”
The Fueter mapping theorem in integral form and the ℱ‐functional calculus (2010, background): “In fact, if two slice monogenic functions f , g coincide on U∩LI for some I∈S, the Identity principle, see [6], guarantees that they coincide on U⊆Rn+1 only when U intersects the real axis.”
Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “Slice hyperholomorphic functions have two main formulations according to the fact that the functions we consider are defined on quaternions and are quaternion-valued, in this case the functions are called slice regular, see [32, 15, 19] or the functions are defined on the Euclidean space R and have ”
The algebra of slice functions (2015, background): “The article [13] addressed the question for the Clifford algebra R3 and [5] treated the case of functions from R m+1 to the Clifford algebra Rm, defining the notion of slice monogenic function.”
Continuous slice functional calculus in quaternionic Hilbert spaces (2012, background): “Continuous slice functional calculus for normal operators 59 References 69”
Extension results for slice regular functions of a quaternionic variable (2009, background): “It is worth to point out that an analogous theory can be developed for functions with values in a Clifford algebra which are called slice monogenic, see [8].”
Entire slice regular functions (2015, methodology): “The theory was developed also for functions with values in a Clifford algebra, see [72], [73], for functions with values in a real alternative algebra, see [117].”
The inverse Fueter mapping theorem (2011, background): “First of all let us recall that slice monogenic functions admit power series expansions only at points on the real axis, see [10].”
Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting (2011, background): “is theory of functions are [56, 33, 38]. For the generalization to functions with values in a Cliﬀord algebra, called again slice hyperholomorphic or slice monogenic functions, we refer the reader to [44, 46, 45], and to [57] for functions with values in a real alternative algebra. Finally, we menti”
A new functional calculus for noncommuting operators (2007, methodology): “Specifically we replace the use of monogenic functions with the new concept, see [2], of slice-monogenic functions.”
The C-property for slice regular functions and applications to the Bergman space (2013, background): “After their definition in [15] which has been generalized to the case of functions with values in a Clifford algebra in [8], these functions have been intensively studied during the past years and their theory, as well as the application to the functional calculus, is treated in the book [9].”
An extension theorem for slice monogenic functions and some of its consequences (2010, methodology): “12 in [6] this series expansion can be extended to a ball B(y0, rI) in R.”
Regular Functions of a Quaternionic Variable (2013, unknown): “The case of functions defined on Rm+1 with values in the Clifford algebra Cl(0,m) has been considered by Fabrizio Colombo, Irene Sabadini and Struppa in [14, 15], where the notion of slice monogenic function is introduced.”