By Miller G. A.

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2a, it suffices to estimate P(a; 2r) lID9! IIp(0;,) for each pair a,r such that X. Then, by ii), we see that v, ID9f(1')l If IIP(b;r) Vt llflIP(a;2r) ror each b e P(a; r), and it follows that lID9! IIP(o;r) II! IIP(o. 3, we want to show that a function is holomorphic 1ff it can be represented locally by its Taylor series (Weierstrass's definition of holomorphic functions). 6 Remark. 6 Prop. 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.

Hint: assume that 0 E then there exist a1,b,c R such that <0; use the function f(z) E. 12g. 9). (a1 — lb1):1 I <1z2 I < 0 1 } is a domain of holomorphy (hint: (X) for 0€ R; for E. 62 b). § 12 A Supplement: Further Extension Theorems for Holomorphic Functions Here we discuss some more consequences of Thullen's Lemma; Proposition 12 A. 3 can serve as motivation for sections 13 and 14. We understand a complex disk D in X a CN to be a set of the form with and din Ca; we also write bd D = d + S' For X sion of ba IR denote the (uniquely determined [Co X.

Ia'I. //) If a E x 0 C", we have that then, with the canonical projection pr: C" —. a multi-index pEN1 can be determined so that

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