Title: Remarks on the Analytic Implicit Function Theorem
Abstract: 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 Bürmann–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.