Functions of One Complex Variable II

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Springer Science & Business Media, Jun 13, 1996 - Mathematics - 396 pages
This is the sequel to my book Functions of One Complex Variable I, and probably a good opportunity to express my appreciation to the mathemat ical community for its reception of that work. In retrospect, writing that book was a crazy venture. As a graduate student I had had one of the worst learning experiences of my career when I took complex analysis; a truly bad teacher. As a non-tenured assistant professor, the department allowed me to teach the graduate course in complex analysis. They thought I knew the material; I wanted to learn it. I adopted a standard text and shortly after beginning to prepare my lectures I became dissatisfied. All the books in print had virtues; but I was educated as a modern analyst, not a classical one, and they failed to satisfy me. This set a pattern for me in learning new mathematics after I had become a mathematician. Some topics I found satisfactorily treated in some sources; some I read in many books and then recast in my own style. There is also the matter of philosophy and point of view. Going from a certain mathematical vantage point to another is thought by many as being independent of the path; certainly true if your only objective is getting there. But getting there is often half the fun and often there is twice the value in the journey if the path is properly chosen.
 

Contents

Return to Basics
1
Runges Theorem
8
Conformal Equivalence for Simply Connected Regions
29
Elementary Properties and Examples of Analytic Functions
69
123
71
Conformal Equivalence with a Circular Region
100
Analytic Covering Spaces
109
The Modular Function
116
Harmonic Functions
206
Entire Functions
221
Hardy Spaces on the Disk
269
Potential Theory in the Plane
301
Singularities
322
The Maximum Modulus Theorem
337
11
355
13
362

5
125
De Brangess Proof of the Bieberbach Conjecture
133
Complex Integration
134
Some Fundamental Concepts from Analysis
169
Harmonic Functions Redux
205

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