Analytic Functions of Several Complex Variables

The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.