A Note on the Order of Contact between Sets in the Complex Plane

• Published in: Journal of mathematical analysis and applications 1998-01, Vol.217 (2), p.707-723
• Main Authors: Spijker, M.N, Straetemans, F.A.J
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Several complex variables and analytic spaces
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1997.5760

This paper deals with the order of contact between arbitrary setsXandYlying in the complex plane C. Let ϕ be a rational function, and ξ0∈C. Suppose, at all pointsxi∈C∪{∞} with ϕ(xi)=ξ0, the setsX,Yhave contact of a given orderpi. In the stability analysis of numerical methods for solving differential equations, the problem arises to determine the order of contact, sayp, between ϕ(X) and ϕ(Y) at ξ0. In this paper a theorem is given according to whichp=maxipi/li, whereli≥1 denotes the order of a certain nonvanishing derivative related to ϕ andxi. The theorem is valid under the assumption that ϕ−1(ϕ(X))⊂cl(X) or ϕ−1(ϕ(Y))⊂cl(Y). This assumption is satisfied in the stability analysis just mentioned, so that the theorem settles the problem arising in that context. The paper also deals with various natural questions related to the specific concept of order (of contact) used in the above mentioned theorem. Some of these questions are important in view of applications.