AMS Graduate Studies in Mathematics 40, 1997

"First of all, we have developed the idea that an introductory book on this subject should emphasize how complex analysis is a natural outgrowth of multivariable real calculus. Complex function theory has, of course, long been an independently flourishing field. but the easiest path into the subject is to observe how at least its rudiments arise directly from ideas about calculus with which the student will already be familiar. We pursue this point of view both by comparing and by contrasting complex variable theory with real-variable calculus.

Second, we have made a systematic attempt to separate analytical ideas, belonging to complex analysis in the strictest sense, from topological considerations. Historically, complex analysis and topology grew up together in the late nineteenth century. And, long ago, it was natural to write complex analysis texts that were a simultaneous introduction to both subjects. But topology has been an independent discipline for almost a century, and it seems to us only a confusion of issues to treat complex analysis as a justification for an introduction to the topology of the plane."

# Chapter 1 : Fundamental ConceptsEdit

# Chapter 2 : Complex Line IntegralsEdit

# Chapter 3 : Applications of the Cauchy IntegralEdit

# Chapter 4 : Meromorphic Functions and ResiduesEdit

# Chapter 5 : The Zeroes of a Holomorphic FunctionEdit

## Section 5.1 : Counting Zeroes and PolesEdit

## Section 5.2 : The Local Geometry of Holomorphic FunctionsEdit

## Section 5.3 : Further Results on the Zeros of Holomorphic FunctionsEdit

## Section 5.4 : The Maximum Modulus PrincipleEdit

## Section 5.5 : The Schwarz LemmaEdit

# Chapter 6 : Holomorphic Functions as Geometric MappingsEdit

# Chapter 7 : Harmonic FunctionsEdit

# Chapter 8 : Infinite Series and ProductsEdit

# Chapter 9 : Applications of Infinite Sums and ProductsEdit

# Chapter 10 : Analytic ContinuationEdit

## Section 10.1 : Definition of an Analytic Function ElementEdit

- I've found the presentation of this construction confusing here and elsewhere, because the equivalence relation tricks me into thinking that something different is about to happen. It seems helpful to outline what we're going to do before we do it. A
*function element*is a pair (f, U) where U is some disc in the complex plane and f is a holomorphic function with domain U -- equivalently, we can think of a power series and the disc on which it converges. (This is also called a*germ*.) Two function elements (f, U) and (g, V) are*direct continuations*of each other if U and V have nonempty intersection and f and g agree on this intersection (so that there is a holomorphic function on the union which agrees with both f and g where defined). A particular function element (f, U) generates a subset F of G, where F is the minimal subset containing (f, U) and containing all direct continuations of its elements.**Note that, in general, this will likely contain function elements (g, V) and (h, V) where g and h are distinct functions.**F is called a*global analytic function,*and its elements are called*branches*of F. With this outline in place, I find that the textbook presentation now makes sense to me.