no. 10
Complex Analysis
A new approach to Cullen-regular functions of a quaternionic variable
[Une nouvelle approche aux fonctions Cullen-régulières d'une variable quaternionelle]

Présenté par : Jean-Pierre Demailly

Gentili, Graziano1 ; Struppa, Daniele C.2
1 Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, Firenze, Italy
2 Department of Mathematics and Computer Sciences, Chapman University, Orange, CA 92866, USA
Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 741-744.

Dans ce travail nous annonçons les éléments et les résultats de base d'une nouvelle théorie des fonctions régulières d'une variable quaternionelle. La théorie que nous décrivons ici s'inspire d'une idée de Cullen, mais nous utilisons une approche plus géométrique pour montrer qu'il est possible de construire une théorie plutôt complète.

In this Note we announce the basic elements and results of a new theory of regular functions of one quaternionic variable. The theory we describe follows an idea of Cullen, but we use a more geometric approach to show that it is possible to build a rather complete theory.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.03.015
Gentili, Graziano 1 ; Struppa, Daniele C. 2

1 Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67 A, Firenze, Italy
2 Department of Mathematics and Computer Sciences, Chapman University, Orange, CA 92866, USA 
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Gentili, Graziano; Struppa, Daniele C. A new approach to Cullen-regular functions of a quaternionic variable. Comptes Rendus. Mathématique, Tome 342 (2006) no. 10, pp. 741-744. doi : 10.1016/j.crma.2006.03.015. http://www.numdam.org/articles/10.1016/j.crma.2006.03.015/

[1] Ahlfors, L. Complex Analysis, McGraw-Hill, New York, 1966

[2] Brackx, F.; Delanghe, R.; Sommen, F. Clifford Analysis, Pitman Res. Notes in Math., vol. 76, 1982

[3] Colombo, F.; Sabadini, I.; Sommen, F.; Struppa, D.C. Analysis of Dirac Systems and Computational Algebra, Birkhäuser, 2004

[4] Cullen, C.G. An integral theorem for analytic intrinsic functions on quaternions, Duke Math. J., Volume 32 (1965), pp. 139-148

[5] Fueter, R. Die Funktionentheorie der Differentialgleichungen Δu=0 und ΔΔu=0 mit vier reellen Variablen, Comment. Math. Helv., Volume 7 (1934), pp. 307-330

[6] Fueter, R. Über einen Hartogs'schen Satz, Comment. Math. Helv., Volume 12 (1939), pp. 75-80

[7] G. Gentili, D.C. Struppa, preprint, 2005

[8] Laville, G.; Ramadanoff, I. Holomorphic Cliffordian functions, Adv. Appl. Clifford Algebras, Volume 8 (1998), pp. 323-340

[9] Laville, G.; Ramadanoff, I. Elliptic Cliffordian functions, Complex Variables Theory Appl., Volume 45 (2001), pp. 297-318

[10] Leutwiler, H. Modified quaternionic analysis in R3, Complex Variables Theory Appl., Volume 20 (1992), pp. 19-51

[11] Rinehart, R.F. Elements of a theory of intrinsic functions on algebras, Duke Math. J., Volume 27 (1960), pp. 1-19

[12] Sudbery, A. Quaternionic analysis, Math. Proc. Cambridge Philos. Soc., Volume 85 (1979), pp. 199-225

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