Remarks on the analytic implicit function theorem (Q1359626)

General concepts from differential calculus in Banach spaces and the corresponding abstract notion of holomorphy have been proved to be useful even in finite-dimensional situations. A classical example in this direction is Rosenbloom's proof of an extended version of the Cauchy-Kovalewski theorem [\textit{P. C. Rosenbloom}, Proc. Sympos. Pure Math. 4, 51-72 (1961; Zbl 0178.44801)]. The author focuses 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. The main purpose of the paper 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.