Comptes Rendus
Complex Analysis
A differential criterium for regularity of quaternionic functions
Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 89-92.

Let Ω 2 . We prove that there exist differential operators T and N, with complex coefficients, such that a function f:Ω ¯ of class C1 is regular if and only if (NjT)f=0 on Ω (j a basic quaternion) and f is harmonic on Ω. At the same time we generalize a result of Kytmanov and Aizenberg. We show that a complex harmonic function h on Ω (Ω connected) is holomorphic if and only if ¯ n h= aL (h) ¯ on Ω, where ¯ n is the normal part of ¯, L is a tangential Cauchy–Riemann operator and a.

Soit Ω 2 . Nous montrons l'existence de deux opérateurs différentiels T et N, à coefficients complexes, telle que une fonction f:Ω de classe C1 est régulière si et seulement si (NjT)f=0 sur Ω (j un quaternion de base de ) et f est harmonique. Nous obtenons aussi une généralisation d'un résultat de Kytmanov et Aizenberg. Nous montrons qu'une fonction harmonique complexe h sur Ω (Ω connexe) est holomorphe si et seulement si ¯ n h= aL (h) ¯ sur Ω, où ¯ n est la composante normale de ¯, L est un opérateur différentiel tangentiel de Cauchy–Riemann et a.

Published online:
DOI: 10.1016/S1631-073X(03)00284-X

Alessandro Perotti 1

1 Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy
     author = {Alessandro Perotti},
     title = {A differential criterium for regularity of quaternionic functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {89--92},
     publisher = {Elsevier},
     volume = {337},
     number = {2},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00284-X},
     language = {en},
AU  - Alessandro Perotti
TI  - A differential criterium for regularity of quaternionic functions
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 89
EP  - 92
VL  - 337
IS  - 2
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00284-X
LA  - en
ID  - CRMATH_2003__337_2_89_0
ER  - 
%0 Journal Article
%A Alessandro Perotti
%T A differential criterium for regularity of quaternionic functions
%J Comptes Rendus. Mathématique
%D 2003
%P 89-92
%V 337
%N 2
%I Elsevier
%R 10.1016/S1631-073X(03)00284-X
%G en
%F CRMATH_2003__337_2_89_0
Alessandro Perotti. A differential criterium for regularity of quaternionic functions. Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 89-92. doi : 10.1016/S1631-073X(03)00284-X.

[1] A.M. Kytmanov The Bochner–Martinelli Integral and its Applications, Birkhäuser, Basel, 1995

[2] A.M. Kytmanov Some differential criteria for the holomorphy of functions in n , Some Problems of Multidimensional Complex Analysis, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Fiz., Krasnoyarsk, 1980, pp. 51-64 263–264 (in Russian)

[3] A.M. Kytmanov; L.A. Aizenberg The holomorphy of continuous functions that are representable by the Bochner–Martinelli integral, Izv. Akad. Nauk Armyan. SSR, Volume 13 (1978), pp. 158-169 (in Russian)

[4] I.M. Mitelman; M.V. Shapiro Differentiation of the Martinelli–Bochner integrals and the notion of hyperderivability, Math. Nachr., Volume 172 (1995), pp. 211-238

[5] M. Naser Hyperholomorphe Funktionen, Sibirsk. Mat. Zh., Volume 12 (1971), pp. 1327-1340 (in Russian); English translation: Siberian Math. J., 12, 1971, pp. 959-968 (1972)

[6] K. Nōno α-hyperholomorphic function theory, Bull. Fukuoka Univ. Ed. III, Volume 35 (1985), pp. 11-17

[7] K. Nōno Characterization of domains of holomorphy by the existence of hyper-conjugate harmonic functions, Rev. Roumaine Math. Pures Appl., Volume 31 (1986) no. 2, pp. 159-161

[8] M.V. Shapiro; N.L. Vasilevski Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-hyperholomorphic function theory, Complex Variables Theory Appl., Volume 27 (1995) no. 1, pp. 17-46

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

[10] N.L. Vasilevski; M.V. Shapiro Some questions of hypercomplex analysis, Complex Analysis and Applications '87 (Varna, 1987), Publ. House Bulgar. Acad. Sciences, Sofia, 1989, pp. 523-531

Cited by Sources:

Comments - Policy