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**Extra resources for Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory**

**Sample text**

For it eCu. see E. II a. 311) vl since T is a compact set on which the geometric series converges absolutely and uniformly. Thus we may interchange summation and integration. • is called an entire (holomorphic)function. Prove Liouville's Theorem: E. 4b. A function fE Every bounded entire function is constant. E. 4c. Let I be an entire function, and suppose that there exist a multi-index v and a constant Show that f is a polynomial of degree at most y >0 such that lf(z)l ylz'l for every z e lvi.

X, Ctm), the set W: is called an analytic polyhedron in X if W is a relatively compact subset of U. In U = 0, is holomorphically convex: since W that case, we deduce from v) that W it follows that is proper. 8 v), the mapping The set U in the definition of W ensures that each union of connected components of W is also an analytic polyhedron if it is at a positive distance from each other connected component of W. In this book we apply only those analytic polyhedra for which Z is a polydisk. vii) Convex domains in (in particular, balls are domains of holomorphy (see E.

9]. Function Theory on Domains in 14 ii) Jf,. iii) f, and converges in I converges in (P(X) to a limit that is independent of the order of summation. 2Q. 7 Abel's Lemma. If a,zv is a power series and we C", then the following statements hold' 0). 1) and If the set {a0w'; VE Ia,z' c C is bounded, then converges in converges in ii) Other;i'ise, the series I I z' and Ia, 1" diverge for every z E such that ;%' E P(z). <1 for every j; thus, by E. 4a, for each Proof 1) For 2 e P(w), we have that I £ > 0, there exists a k €N such that I la,z'I= IyIk where C:=SUpIa,W"I.