By Raghavan Narasimhan

The unique version of this ebook has been out of print for a few years. The look ance of the current moment variation owes a lot to the initiative of Yves Nievergelt at japanese Washington collage, and the help of Ann Kostant, arithmetic Editor at Birkhauser. because the booklet was once first released, a number of humans have remarked at the absence of routines and expressed the opinion that the ebook might were extra invaluable had routines been integrated. In 1997, Yves Nievergelt expert me that, for a decade, he had usually taught a path at jap Washington in keeping with the ebook, and that he had systematically compiled workouts for his path. He kindly placed his paintings at my disposal. therefore, the current version seems in components. the 1st is largely only a reprint of the unique variation. i've got corrected the misprints of which i've got turn into conscious (including these mentioned to me by way of others), and feature made a small variety of different minor adjustments.

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Now 1 z-w )-1 1 ( w-a 1--z-a ;:-a --=-- = L 00 n=O (w-ar n+l (z-a) ' the series converging uniformly for Iz - al = p if Iw - al < p (w is fixed). Hence, for wED(a,p), we have 00 f(w)=Lcn(w-a)n n=O 1 S fez) wherecn =-2· ( r+ 1dz . nl y z-a It follows from Lemma 1 that en = pn)(a)jn1. Thus, the Taylor series off at a converges to f on D(a, p). Since p< r is arbitrary, this proves the result. Corollary. Let Q be open in

1 The sequence {fqJ then converges uniformly on This proves Montel's theorem. =1 y , hence on K. As an instance of the way in which Montel's theorem may be applied, we prove the following classical result (which is, in fact, equivalent to Montel's theorem). Theorem 3. (Vitali) Let D be a connected open set in CC and let A cD be a subset which has at least one point of accumulation in D. }n~ 1 be a sequence of functions holomorphic on D which is uniformly bounded on any compact subset of D. (a)} converges for all aEA as n-+oo.

Proof. Sulficiency 01 the condition. If II U - {a} gEYf(U) with gIU-{a}=IIV-{a}, and we have 1f =g 1S bounded, there 1S on U - {a} . If I/(z)l-oo as z-a, z#O, there is a disc D(a,p)=VcU such that I/(z) I ~ 1 for z E V - {a}. Again, by Theorem 2, there is hE Yf(V) such that hlV - {a} = IIIIV - {a}. We then have h f = 1 on V - {a} . Thus I is meromorphic. Necessity 01 the condition. Let a EE, V=D(a, p) a disc around a with V ( l E = {a} on which there exist h, g E Yf (V), h ¥= such that hi = g on V- {a}.