Few other books purport to be a second course in complex analysis. This book differs in that it covers more modern topics and is more geometric in focus. Most texts on complex variable theory contain the same material. However, complex analysis is a vast and diverse subject with a long history and many aspects. A second course will benefit students and introduce these new topics that they might not otherwise experience.

Lars Ahlfors alone invented many new parts of the subject; Lipman Bers made decisive contributions, and there are many others. It is easy to justify a “second course” in complex analysis. That is what this book purports to be.

Some of the topics presented here are:

• harmonic measure

• extremal length

• Riemann surfaces

• uniformization

• automorphism groups

• the Schwarz lemma and its generalizations

• analytic capacity

• the Bergman theory

• invariant metrics

• Picard’s theorem

• the boundary Schwarz lemma

The goal is to expose the reader to unfamiliar parts of the subject of complex variables and perhaps to pique interest in further reading. As with the authors’ other books, not only theorems and proofs are included, but also many examples and some exercises. Numerous graphics illustrate the key ideas.