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**Extra resources for From Holomorphic Functions to Complex Manifolds**

**Example text**

For v = 1 , . . ' n - 1 let Tv be the automorphism of D defined by = := Then h : pn ---+ pn with h(z1 , . . , zn ) := (T1 (z2) , . . , Tn - 1 ( zn ) , z! ) is holo morphic, h(O) (z�, O) , and h({z E pn : ! z1 l > q}) C {w E pn : lwn l > q } . 3. Extending a holomorphic function across a hypersurface 1. We define q1 := The Continuity Theorem 47 q, and for v = 2, . . , n choose qv such that h (D X Dq2 (0) X · · · X Dq, (O)) C U' X D. 2, . . 3). Since P n G' G' is connected, the proposition follows from the continuity theorem.

Let f(z) = :Z::: v >O avz v and g (z) = l: v >O bv z v be two convergent power series in a neighborhood U = U(O) c en . Ifthere is a neigborhood V(O) c U with f l v = 9 l v, then a, = b, fo r all v . PROOF: We know that f and g are holomorphic. Then D v f (0) = D v g ( O ) for all v , and successive differentiation gives Dv f ( O )= v! av and Dv g ( O ) = v! b,. • 4 . 8 Cor ollary. Let G c e n be a domain, Zo E G a point, and f : B --+ e a holomorphic fu nction. If f ( z) = :Z::: v >o av (z - zo) v is the (uniquely deter m ined) power series expansion near z0 -E G, then a, = � .

D.. " (zo) t , := w . (D.. ' (z o)) t + w · (D.. '' (zo)) t . " (zo ) ll , it follows that f is differentiable at z0 with derivative L. • I. Holomorphic Functions 28 Wir tin ger ' s Calculus Definition . Let f : B representation --+ e be real differentiable at zo . If we have a f(z) = f(zo) + (z - zo) . �' (z) t + (z - zo) . �" (z) t , with �' and � " continuous at z0, then the uniquely determined numbers and of Ozv (zo) = fzv (zo) := ev - �"(zo) t are called the Wi rtinger derivatives of f at zo.