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.
KeywordsHolomorphic Function Compact Support Meromorphic Function Open Covering Principal Part
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