# Download e-book for kindle: A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag

By Wilhelm Schlag

ISBN-10: 0821898477

ISBN-13: 9780821898475

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Additional info for A Course in Complex Analysis and Riemann Surfaces

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Proof. Let U c n be simply-connected, say a disk. By Proposition 1 . ) where f E 1i(U) . Since f E C00 (U) , so is u. Moreover, f (zo) = 1 . rt j fr 1 f (z) dz = f f (zo + re 27rit ) dt. 20) . For the maximum principle, suppose that u attains a local extremum on some disk in n. Then it follows from ( 1 . 20) that u has to be constant on that disk. Since any two points in n are contained in a simply-connected subregion of n, we conclude from the existence of conjugate harmonic functions on simply-connected regions as well as the uniqueness theorem for analytic functions that u is globally D constant.

7. Applications of Cauchy's theorems It also implies the astonishing fact that holomorphic functions are in fact analytic. This is done by noting that the integrand in (l. 14) is analytic relative to In other words, we reduce ourselves to the geometric series. z. In fact, every f E 1i(O) is represented by a convergent power series on D(zo,r) where r = dist (zo , 80) . Corollary 1 . 2 1 . A (O) = 1i(O) . Proof. We proved in Lemma 1. 3 that analytic functions are holomorphic. For the converse, we use the previous proposition to conclude that with 'Y 24 1.

Indeed, each of the roots of f(zo) (which are all distinct ) has a unique pre-image under cp(z). In summary, f has the stated n-to-one mapping property. The openness is now also evident. D Figure 1 . 7 shows the case of 6. The dots symbolize the six pre­ images of some point. We remark that any point zo E n for which � 2 is called a branch point. The branch points are precisely the zeros of f' in n and therefore form a discrete subset of n ( this means that every point of this subset is isolated from the other ones ) .