Complex Analysis
A differential criterium for regularity of quaternionic functions
[Une condition différentielle de régularité pour les fonctions quaternioniennes]
Comptes Rendus. Mathématique, Tome 337 (2003) no. 2, pp. 89-92.

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.

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.

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DOI : 10.1016/S1631-073X(03)00284-X
Perotti, Alessandro 1

1 Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy
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Perotti, Alessandro. A differential criterium for regularity of quaternionic functions. Comptes Rendus. Mathématique, Tome 337 (2003) no. 2, pp. 89-92. doi : 10.1016/S1631-073X(03)00284-X. http://www.numdam.org/articles/10.1016/S1631-073X(03)00284-X/

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