Methods in mathematical logic. Proc. of 6th Latin American by Carlos A. Di Prisco

By Carlos A. Di Prisco

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Extra info for Methods in mathematical logic. Proc. of 6th Latin American Symposium on Mathematical Logic, Caracas, Venezuela, 1983

Example 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.

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