Residues and D-modules

The work Mémoire sur une propriété générale d’une classe très étendue de fonctions trancendantes by Niels Henrik Abel started a new era where geometry, algebra and complex analysis are brought together. It is remarkable that Abel already in 1826 described the process of integrating algebraic functions over cycles — and more generally over chains - which eliminate some variables while the function of the remaining variables has a specific transcendental nature. For example, the conclusive result from 1965 by N. Nilsson in [81] was already suggested in Abel’s work. Multidimensional residue theory is a subject which also has developed in the spirit of Abel. This article describes how regular holonomic D x-modules can be realised by distributions satisfying regular holonomic systems on a complex manifold X. Such distributions emerge from Nilsson class functions, i.e., multi-valued analytic functions defined in the complement of a hypersurface T with finite determination and moderate growth along T. Residues and D-modules are treated in Sects. 2-6. Section 1 describes historic examples, foremost related to work by Niels Henrik Abel. Let us finish by citing C. H. Hermite from [9, p. 8]: Abel avait fait la decouverte capitale qu’une somme d’un nombre quelconque d’integrales à limites arbitraires, de la mème fonction algébrique, s’exprime par un nombre fixe d’integrales semblables auxquelles s’ajoute une quantite algébrique et logarithmique.