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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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A Course in Complex Analysis and Riemann Surfaces (Hardcover)
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A Course in Complex Analysis and Riemann Surfaces (Hardcover)
Series: Graduate Studies in Mathematics
Expected to ship within 12 - 17 working days
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Complex analysis is a cornerstone of mathematics, making it an
essential element of any area of study in graduate mathematics.
Schlag's treatment of the subject emphasizes the intuitive
geometric underpinnings of elementary complex analysis that
naturally lead to the theory of Riemann surfaces. The book begins
with an exposition of the basic theory of holomorphic functions of
one complex variable. The first two chapters constitute a fairly
rapid, but comprehensive course in complex analysis. The third
chapter is devoted to the study of harmonic functions on the disk
and the half-plane, with an emphasis on the Dirichlet problem.
Starting with the fourth chapter, the theory of Riemann surfaces is
developed in some detail and with complete rigor. From the
beginning, the geometric aspects are emphasized and classical
topics such as elliptic functions and elliptic integrals are
presented as illustrations of the abstract theory. The special role
of compact Riemann surfaces is explained, and their connection with
algebraic equations is established. The book concludes with three
chapters devoted to three major results: the Hodge decomposition
theorem, the Riemann-Roch theorem, and the uniformization theorem.
These chapters present the core technical apparatus of Riemann
surface theory at this level. This text is intended as a fairly
detailed, yet fast-paced intermediate introduction to those parts
of the theory of one complex variable that seem most useful in
other areas of mathematics, including geometric group theory,
dynamics, algebraic geometry, number theory, and functional
analysis. More than seventy figures serve to illustrate concepts
and ideas, and the many problems at the end of each chapter give
the reader ample opportunity for practice and independent study.
General
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