Riv. Mat. Univ. Parma, Vol. 11, No. 1, 2020

Graziano Gentili [a], Giulia Sarfatti [b] and Daniele C. Struppa [c]

A family of Cauchy-Riemann type operators

Pages: 123-138
Received: 31 January 2019
Accepted: 4 March 2019
Mathematics Subject Classification (2010): 30A, 30-01.
Keywords: General properties of holomorphic functions.
[a]: Dipartimento di Matematica e Informatica ''U. Dini'', Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
[b]: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy
[c]: The Donald Bren Presidential Chair in Mathematics, Chapman University, One University Drive, Orange, CA 92866

Abstract: A natural question is whether and in which sense the definition of a holomorphic function depends on the choice of the two vectors $$\{1, i\}$$ that form a basis of $$\mathbb{C}$$ over $$\mathbb{R}$$. In fact these two vectors determine both the form of the Cauchy-Riemann operator, and the splitting of a holomorphic function in its harmonic real and imaginary components. In this paper we consider the basis $$\{1, e^{i\theta}\}$$ of $$\mathbb{C}$$ over $$\mathbb{R}$$, and define as $$\theta$$-holomorphic the functions that belong to the kernel of a Cauchy-Riemann type operator determined by this basis. We study properties of these functions, and discuss the relation between them and classical holomorphic functions. This analysis will lead us to discover the special role that $$\theta=\pi/2$$ plays, that renders the theory of holomorphic functions special among this family of theories.

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