Circular centered forms for rational functions in several complex variables
This paper continues the second author's work in complex interval analysis. [See \textit{J. Rokne}, J. Comput. Appl. Math. 17, 309-327 (1987; Zbl 0614.65051), and \textit{P. Bao, J. Rokne}, Acta Appl. Math. 16, No. 3, 261-280 (1989; Zbl 0687.65018)]. Let \(f: \mathbb{C}^ n \to \mathbb{C}\) be a rational function of \(n\) complex variables. Let \(Z = \langle c,r\rangle\) denote a disk-shaped subset in \(\mathbb{C}^ n\), characterized by the ``center'' \(c\in \mathbb{C}^ n\) and the ``radius'' \(r \in \mathbb{R}^ n\). Define \(f(Z):=\{f(z)| z\in Z\}\). One basic aim of interval analysis is the construction of a disk \(D\) in \(\mathbb{C}\) that contains the set \(f(Z)\). The authors present explicit formulas for such disks \(D\), in particular they provide ``centered forms'', where the center of \(D\) is \(f(c)\). Furthermore, the authors prove interesting convergence properties of their centered forms. The paper is purely theoretical, no numerical results are reported.
scientific article; zbMATH DE number 69598