Abstract :
This paper deals with the order of contact between arbitrary sets X and Y lying
in the complex plane C. Let w be a rational function, and j 0gC. Suppose, at all
points xigCj `4 with w xi.sj 0 , the sets X, Y have contact of a given order
pi. In the stability analysis of numerical methods for solving differential equations,
the problem arises to determine the order of contact, say p, between w X. and
w Y. at j 0. In this paper a theorem is given according to which psmaxi pirli,
where liG1 denotes the order of a certain nonvanishing derivative related to w
and xi. The theorem is valid under the assumption thatwy1 w X..;cl X. or
wy1 w 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.
