The Fundamental Theorem in Complex Function Theory
This introductory chapter is meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z. In the first section we “solve” two natural problems using complex analysis. In the second, we state what w
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The Fundamental Theorem in Complex Function Theory
This introductory chapter is meant to convey the need for and the intrinsic beauty found in passing from a real variable x to a complex variable z. In the first section we “solve” two natural problems using complex analysis. In the second, we state what we regard as the most important result in the theory of functions of one complex variable, which we label the fundamental theorem of complex function theory, in a form suggested by the teaching and exposition style of Lipman Bers; its proof will occupy most of this volume. The next two sections of this chapter include an outline of our plan for the proof and an outline for the text, respectively; in subsequent chapters we will define all the concepts encountered in the statement of the theorem in this chapter. The reader may not be able at this point to understand all (or any) of the statements in our fundamental theorem or to appreciate its depth and might choose initially to skim this material. All readers should periodically, throughout their journey through this book, return to this chapter, particularly to the last section, that contains an interesting account of part of the history of the subject.
1.1 Some Motivation 1.1.1 Where Do Series Converge? In the calculus of a real variable one encounters two series that converge for jxj < 1 but in no larger open interval (on the real axis): 1 D 1 x C x 2 C .1/n x n C 1Cx and 1 D 1 x 2 C x 4 C .1/n x 2n C : 1 C x2 R.E. Rodr´ıguez et al., Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics 245, DOI 10.1007/978-1-4419-7323-8 1, © Springer Science+Business Media New York 2013
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1 The Fundamental Theorem in Complex Function Theory
It is natural to ask why these two series that are centered at the origin have radius 1 of convergence 1. The answer for the first one is natural: the function 1Cx has a singularity at x D 1, and so the series certainly cannot represent the function at this point, which is at distance 1 from 0. For the second series, the answer does 1 not appear readily within real analysis. However, if we view 1Cx 2 as a function of 1 the complex variable x, then we again conclude that the series representing this function should have radius of convergence 1, since that is the distance from 0 to the singularities of the function; they are at ˙{.
1.1.2 A Problem on Partitions A natural question in elementary additive number theory is the following: Is it possible to partition the positive integers Z>0 into finitely many (more than one) infinite arithmetic progressions with distinct differences? The answer is no. It is obviously possible to construct such partitions if some differences are allowed to be equal. So assume that the differences are different and to the contrary that Z>0 D S1 [ S2 [ [ Sn ; where n 2 Z>1 , and for 1 i n, Si is an arithmetic progression with initial term ai and difference di , for 1 i < j n, Si \ Sj D ;, and 1 < d1 < d2 < < dn . Then 1 X X X X zi
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