Multidimensional Integral Representations

Abstract

The theory of integral representations is important in multidimensional complex analysis. It continues to develop rapidly and is finding new applications in multidimensional complex analysis, as well as in other areas of mathematics [see, for example, monographs Aizenberg and Yuzhakov (Integral Representations and Residues in Multidimensional Complex Analysis. AMS, Providence, 1983), Khenkin (Several Complex Variables I. Encyclopedia of Mathematical Sciences, vol. 7, pp. 19–116. Springer, New York, 1990), Krantz (Function Theory of Several Complex Variables, 2nd edn. Wadsworth & Brooks/Cole, Pacific Grove, 1992), Kytmanov (The Bochner–Martilnelli Integral and Its Applications. Birkhauser Verlag, Basel, 1995), Rudin (Function Theory in the Unit Ball of \(\mathbb{C}^{n}\). Springer, New York, 1980), Shabat (Introduction to Complex Analysis. Part 2: Functions of Several Complex Variables. AMS, Providence, 1992), Vladimirov (Methods of the Theory of Functions of Many Complex Variables. MIT Press, Cambridge, 1966)]. This chapter provides those integral representations, which are then used in other chapters. Of course, we do not have space to mention all integral formulas known at this time. We leave out of the scope of this book the formulas of integration by manifolds of smaller dimension (such as the multiple Cauchy formula). The theory of multidimensional residues will be used just a little in the final chapters. We will only dwell on the formulas where integration is performed over the entire boundary of domain. The presentation is designed to show the logic of proceeding from the classical Bochner–Green formula to the Khenkin–Ramirez formula that has found a number of important applications in multidimensional complex analysis.