AMS Graduate Studies in Mathematics 40, 1997

Complex Analysis

"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.

Chapter 11 : TopologyEdit

Chapter 12 : Rational Approximation TheoryEdit

Chapter 13 : Special Classes of Holomorphic FunctionsEdit

Chapter 14 : Hilbert Spaces of Holomorphic Functions, the Bergman Kernel, and Biholomorphic MappingsEdit

Chapter 15 : Special FunctionsEdit

Chapter 16 : The Prime Number TheoremEdit

Appendix A : Real AnalysisEdit

Appendix B : The Statement and Proof of Goursat's TheoremEdit