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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
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