Functions of Several Complex Variables

At a naive level, the analysis of several real variables is much like the real analysis of one variable; the main difference is that one deals with n-tuples of reals instead of scalars and one needs matrices to keep track of information. Of course deeper study reveals much complexity and ri chness in analysis of several real variables. It is noteworthy that, for the most part, this richness was discovered rather late in the history of the subject—mostly in the last fifty years.

The history of analysis of several complex variables is quite different. Early in the subject, in the first decade of the twentieth century, two remarkable discoveries indicated that this area has an incredible depth and variety which one complex variable does not even hint at. Even today we have only scratched the surface of several complex variables.

Let us briefly discuss the two developments which established several complex variables as a subject in its own right. The first is related to the Riemann mapping theorem. As we have discussed throughout the present volume, Riemann's theorem asserts that, with the single exception of the plane, any domain topologically equivalent to the disc is conformally equivalent to it. One might expect an analogue to this result in two complex variables.