Soit . Nous montrons l'existence de deux opérateurs différentiels T et N, à coefficients complexes, telle que une fonction de classe C1 est régulière si et seulement si (N−jT)f=0 sur (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 ( connexe) est holomorphe si et seulement si sur , où est la composante normale de , L est un opérateur différentiel tangentiel de Cauchy–Riemann et .
Let . We prove that there exist differential operators T and N, with complex coefficients, such that a function of class C1 is regular if and only if (N−jT)f=0 on (j a basic quaternion) and f is harmonic on . At the same time we generalize a result of Kytmanov and Aizenberg. We show that a complex harmonic function h on ( connected) is holomorphic if and only if on , where is the normal part of , L is a tangential Cauchy–Riemann operator and .
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@article{CRMATH_2003__337_2_89_0, author = {Perotti, Alessandro}, 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}, url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00284-X/} }
TY - JOUR AU - Perotti, Alessandro 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 UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00284-X/ DO - 10.1016/S1631-073X(03)00284-X LA - en ID - CRMATH_2003__337_2_89_0 ER -
%0 Journal Article %A Perotti, Alessandro %T A differential criterium for regularity of quaternionic functions %J Comptes Rendus. Mathématique %D 2003 %P 89-92 %V 337 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00284-X/ %R 10.1016/S1631-073X(03)00284-X %G en %F CRMATH_2003__337_2_89_0
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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