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Introduction

how one might have found a certain argument, etc. In several places we give

two versions of a proof, one more "abstract" than the other; the reader who

wishes to attain a clear understanding of the difference between the forest

and the trees is encouraged to contemplate both proofs until he or she sees

how they are really just different expositions of the same underlying idea.

Many results in elementary complex analysis (pointwise differentiability

implies smoothness, a uniform limit of holomorphic functions is holomorphic,

etc.) are really quite surprising. Or at least they should be surprising; we

include examples from real analysis for the benefit of readers who might not

otherwise see what the big deal is.

There are a few ways in which the content differs from that of the typical

text. First, the reader will notice an emphasis on (holomorphic) automor-

phism groups and an explicit mention of the notion of covering spaces. These

concepts are used in incidental ways in the first half of the book; for exam-

ple linear-fractional transformations arise naturally as the automorphisms

of the Riemann sphere instead of being introduced as an ad hoc class of

conformal maps in which it just happens that various calculations are easy,

covering maps serve to unify various results on analytic continuation, etc.;

then it turns out that some not-quite-trivial results on automorphisms and

covering maps are crucial to the proof of the Big Picard Theorem.

Probably the most unusual aspect of the content is the inclusion of a

section on the relation between Brownian motion and the Dirichlet problem.

In most of the text we have tried to achieve a fairly high standard of rigor,

but in this section the notion of rigor simply flies out the window: We do not

even include precise definitions of the things we're talking about! We de-

cided to include a discussion of this topic even though we could not possibly

do so rigorously (considering the prerequisites we assume) because Brownian

motion gives the clearest possible intuition concerning the Dirichlet prob-

lem. Readers who are offended by the informal nature of the exposition

in this section are encouraged to think of it not so much as a lecture but

rather a conversation in the departmental lounge or over a few beers on a

Friday afternoon.

Finally, the proof of the Big Picard Theorem will probably be new to

most readers, possibly including many experts. The proof is certainly not

simpler or shorter than the proofs found in typical texts, but it seems very

interesting, at least to me: It proceeds by essentially a direct generalization

of the standard "one-line" proof of the Little Picard Theorem. (See the

discussion of Theorem A and Theorem B in Chapter 20.)

It will be clear to many readers that I first learned much of this material

from [R]. Very few references are given; all of the results are quite standard,

and I doubt that any of the proofs are new. Indeed, for some time I thought