Holomorphy tests based on Cauchy's integral formula
type: Item
instance of: scholarly article
title: Holomorphy tests based on Cauchy's integral formula
zbMATH Open document ID: 0873.30023
author: Daniel Pascuas
author: C. Cascante
published in: Pacific Journal of Mathematics
publication date: 1997-11-02T00:00:00Z
review text: In this paper, \(X\) is either \(\mathbb{C}\), the open unit disk \(\mathbb{D}\) or the open upper half-plane \(\mathbb{U}\). \(G\) denotes the group \(M(2)\) of rigid motions of \(\mathbb{C}\) in the first case and in the other cases the full group of conformal automorphisms of \(X\). Cauchy's integral formula gives for rectifiable loops \(\Gamma\) in \(X\) \[ \int_\Gamma{(f\circ\sigma)(z)\over z-a} {dz\over 2\pi i}=\text{Ind}(\Gamma, a)\cdot(f\circ\sigma)(a)\tag{1} \] for all \(f\in H(X)\), \(\sigma\in G\), \(a\in X\backslash\Gamma\). This paper investigates for what Jordan curves \(\Gamma\) and \(a\in X\backslash\Gamma\) the validity of (1) for all \(\sigma\in G\) entails the holomorphy of \(f\) in \(X\). There are strong analogies with the Morera and Pompeiu problems and techniques used in their study are relevant here too. The authors acknowledge a particular debt to the work of \textit{C. A. Berenstein} and \textit{L. Zalcman} [Comment. Math. Helv. 55, 593-621 (1980; Zbl 0452.43012)] and \textit{C. A. Berenstein} and \textit{D. Pascuas} [Isr. J. Math. 86, No. 1-3, 61-106 (1994; Zbl 0827.30001)]. There are methodological differences between the case \(X=\mathbb{C}\) (which is invariant under \(M(2)\)) and the other cases (which are not invariant). That invariance is somehow reflected in the fact that in the planar case there are points, where (1) is automatically satisfied for certain non-holomorphic functions. For example, if \(\Gamma\) is a circle and \(a\) its center, every harmonic function \(f\) satisfies (1), so a conclusion ``\(f\) holomorphic'' is out of the question. Nevertheless, Theorem 1.1 asserts that if \(D\) is a (Euclidean) disk inside \(X\), \(\Gamma=\partial D\), \(a\in X\backslash\Gamma\) is not the center of \(D\) and \(f\in C(X)\) satisfies (1) for all \(\sigma\in G\), then \(f\) is holomorphic in \(X\). In proving Theorem 1.1 for \(X=\mathbb{U}\) the following mean-valued theorem is obtained. Corollary 1.2: Let \(D\) be a closed (Euclidean…
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 912136
zbMATH Keywords: Morera problem
zbMATH Keywords: Pompeiu problem
zbMATH Keywords: Cauchy's integral
reviewed by: Robert Burckel
MaRDI profile type: Publication
full work available at URL: pjm.1995.171.89
OpenAlex ID: W1557903600
DOI: 10.2140/PJM.1995.171.89
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description: scientific article; zbMATH DE number 912136