Meromorphic functions of several variables

This chapter presents the analogues of the Mittag-Leffler and Weierstrass theorems for functions of several complex variables. To this end it develops fundamental methods of multivariable complex analysis that reach far beyond the applications we are going to give here. –

Meromorphic functions

of several variables are defined as local quotients of holomorphic functions (VI.2); the definition requires some information on zero sets of holomorphic functions (VI.1). After introducing

principal parts

and

divisors

we formulate the main problems that arise:

To find a meromorphic function with i) a given principal part

(first Cousin problem)

ii) a given divisor

(second Cousin problem);

iii) to express a meromorphic function as a quotient of globally defined holomorphic functions

(Poincaré problem). These problems are solved on

polydisks

– bounded or unbounded, in particular on the whole space – in VI.6–8. The essential method is a constructive solution of the

inhomogeneous Cauchy-Riemann equations

(VI.3 and 5) based on the one-dimensional inhomogeneous Cauchy formula – see Chapter IV.2. Along the way, various extension theorems for holomorphic functions are proved (VI.1 and 4). Whereas the first Cousin problem can be completely settled by these methods, the second requires additional topological information which is discussed in VI.7, and for the Poincaré problem one needs some facts on the ring of convergent power series which we only quote in VI.8.