Remarks on the Analytic Implicit Function Theorem*

In this paper, we focus attention on the problem of to what extent abstract 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 obtained 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 version of such a formula. The basic idea of the proof consists in combining the technique of multiple residues with certain elementary concepts multilinear algebra.from differential calculus in Banach spaces