CV3 - Complex Variable Part III T. W. K orner June 9, 1999...

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Unformatted text preview: Complex Variable Part III T. W. K orner June 9, 1999 Small print This is just a first draft of part of the course. The content of the course is what I say, not what these notes say. I should very much appreciate being told of any corrections or possible improvements and might even part with a small reward to the first finder of particular errors. This document is written in L A T E X2e and stored in the file labelled ~twk/FTP/CV3.tex on emu in (I hope) read permitted form. My e-mail address is twk@dpmms . Contents 1 Simple connectedness and the logarithm 2 2 The Riemann mapping theorem 6 3 Normal families 8 4 Proof of the Riemann mapping theorem 11 5 Infinite products 13 6 Fourier analysis on finite Abelian groups 17 7 The Euler-Dirichlet formula 21 8 Analytic continuation of the Dirichlet functions 23 9 L (1 , ) is not zero 25 10 Natural boundaries 26 11 Chebychev and the distribution of primes 28 1 12 The prime number theorem 29 13 Boundary behaviour of conformal maps 33 14 Picards little theorem 38 15 References and further reading 39 1 Simple connectedness and the logarithm This is a second course in complex variable theory with an emphasis on technique rather than theory. None the less I intend to be rigorous and you should feel free to question any hand waving that I indulge in. But where should rigour start? It is neither necessary nor desirable to start by reproving all the results of a first course. Instead I shall proceed on the assumption that all the standard theorems (Cauchys theorem, Taylors theorem, Laurents theorem and so on) have been proved rigourously for analytic functions 1 on an open disc and extend them as necessary. Cambridge students are (or, at least ought to be) already familiar with one sort of extension. Definition 1 An open set U in C is called disconnected if we can find open sets U 1 and U 2 such that (i) U 1 U 2 = U , (ii) U 1 U 2 = , (iii) U 1 ,U 2 6 = . An open set which is not disconnected is called connected . Theorem 2 If U is an open connected set in C and f : U C is analytic and not identically zero then all the zeros of f are isolated that is, given w U with f ( w ) = 0 we can find a > such that D ( w, ) U and f ( z ) 6 = 0 whenever z D ( w, ) and z 6 = w . Here and elsewhere D ( w, ) = { z : | w- z | < } . The hypothesis of connectedness is exactly what we need in Theorem 2. Theorem 3 If U is an open set then U is connected if and only if the zeros of every non-constant analytic function on U are isolated. 1 Analytic functions are sometimes called holomorphic functions. We shall call a func- tion which is analytic except for poles a meromorphic function....
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CV3 - Complex Variable Part III T. W. K orner June 9, 1999...

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