REMARKS ON THE ANALYTIC IMPLICIT FUNCTION THEOREM

In this paper, we focus attention on the problem of to what extent abs tract Banach space concepts can be utilized in order to simplify well- known formulas for the power series expansion of an implicitly defined holomorphic mapping in the finite-dimensional case. The formulas obta ined in the previous literature (roughly speaking, extensions of the B urmann-Lagrange formula) are extremely lengthy and involved. Our main purpose here is to give a more condensed and more comprehensible versi on of such a formula. The basic idea of the proof consists in combinin g the technique of multiple residues with certain elementary concepts (multilinear algebra) from differential calculus in Banach spaces. (C) 1997 Academic Press.