Complex Analysis (Cambridge Mathematical Textbooks)

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Preface
This book provides a graduate-level introduction to complex analysis. There are four points
of view for this subject due primarily to Cauchy, Weierstrass, Riemann and Runge. Cauchy
thought of analytic functions in terms of a complex derivative and through his famous integral
formula. Weierstrass instead stressed the importance of power series expansions. Riemann
viewed analytic functions as locally rigid mappings from one region to another, a more geometric point of view. Runge showed that analytic functions are nothing more than limits of
rational functions. The seminal modern text in this area was written by Ahlfors [1], which
stresses Cauchy’s point of view. Most subsequent texts have followed his lead. One aspect
of the first-year course in complex analysis is that the material has been around so long that
some very slick and elegant proofs have been discovered. The subject is quite beautiful as a
result, but some theorems then may seem mysterious.
I have decided instead to start with Weierstrass’s point of view for local behavior. Cartan [4]
has a similar approach. Power series are elementary and give you many non-trivial functions
immediately. In many cases it is a lot easier to see why certain theorems are true from this
point of view. For example, it is remarkable that a function which has a complex derivative
actually has derivatives of all orders. However, the derivative of a power series is just another
power series and hence has derivatives of all orders.
Cauchy’s theorem is a more global result concerned with integrals of analytic functions.
Why integrals of the form
dz are important in Cauchy’s theorem is very easy to understand using partial fractions for rational functions. So we will use Runge’s point of view for
more global results: analytic functions are simply limits of rational functions.
As a pedagogic device we will use the term “analytic” for local power series expansion
and “holomorphic” for possessing a continuous complex derivative. We will of course prove
that these concepts (and several others) are equivalent eventually, but in the early chapters the
reader should be alert to the different definitions.
The emphasis in Chapters 1–6 is to view analytic functions as behaving like polynomials
or rational functions. Perhaps the most important elementary tool in this subject is the maximum principle, highlighted in Chapter 3. Runge’s theorem is proved in Chapter 4 and is used
to prove Cauchy’s theorem in Chapter 5. Chapter 6 uses color to visualize complex-valued
functions. Given a coloring of the complex plane, a function f can be illustrated by placing
the color of f(z) at the point z. See Section A.2 of the appendix for a computer program to
do this.
Chapters 7 and 8 introduce harmonic and subharmonic functions and highlight their application to the study of analytic functions. Chapter 8 includes a method, called the geodesic
zipper algorithm, for numerically computing conformal maps, which is fast and simple to
program. Together with Harnack’s principle, it is used to give a somewhat constructive proof

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