The material in this chapter intentionally interrupts our theoretical development about Hilbert spaces. There are at least two reasons for this interruption. One reason is that this material is a prerequisite for understanding the Hilbert space A2(Bn) of square-integrable holomorphic functions on the unit ball; this particular Hilbert space arises in a crucial way in the proof of Theorem VII.1.1, the main result in the book. A second reason is the feeling that abstract material doesn't firmly plant itself in one's mind unless it is augmented by concrete material. Conversely the presentation of concrete material benefits from appropriate abstract interludes.

In this chapter we introduce holomorphic functions of several complex variables. This presentation provides only a brief introduction to the subject. Multi-index notation and issues involving calculus of several variables also appear here. Studying them allows us to provide a nice treatment of the gamma and beta functions. We use them to compute the Bergman kernel function for the unit ball, thereby reestablishing contact with Hilbert spaces.

Holomorphic functions

Our study of A2(Bn) requires us to first develop some basic information about holomorphic functions of several complex variables. As in one variable, holomorphic functions of several variables are locally represented by convergent power series. Although some formal aspects of the theories are the same, geometric considerations change considerably in the higher-dimensional theory.

In order to avoid a profusion of indices we first introduce multiindex notation. This notation makes many computations in several variables both easier to perform and simpler to expose.

Suppose z = (z1, …, zn) ∈ Cn, and let (α1, …, αn) be an n-tuple of nonnegative integers.