This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 email 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 EulerDirichlet 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 nonconstant 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....
View Full
Document
 Fall '11
 DrewArmstrong

Click to edit the document details