With this second volume, we enter the intriguing world of complex
analysis. From the first theorems on, the elegance and sweep of the
results is evident. The starting point is the simple idea of extending a
function initially given for real values of the argument to one that is
defined when the argument is complex. From there, one proceeds to the
main properties of holomorphic functions, whose proofs are generally
short and quite illuminating: the Cauchy theorems, residues, analytic
continuation, the argument principle.
With this background, the reader is ready to learn a wealth of
additional material connecting the subject with other areas of
mathematics: the Fourier transform treated by contour integration, the
zeta function and the prime number theorem, and an introduction to
elliptic functions culminating in their application to combinatorics and
number theory.
Thoroughly developing a subject with many ramifications, while striking
a careful balance between conceptual insights and the technical
underpinnings of rigorous analysis, Complex Analysis will be welcomed
by students of mathematics, physics, engineering and other sciences.
The Princeton Lectures in Analysis represents a sustained effort to
introduce the core areas of mathematical analysis while also
illustrating the organic unity between them. Numerous examples and
applications throughout its four planned volumes, of which Complex
Analysis is the second, highlight the far-reaching consequences of
certain ideas in analysis to other fields of mathematics and a variety
of sciences. Stein and Shakarchi move from an introduction addressing
Fourier series and integrals to in-depth considerations of complex
analysis; measure and integration theory, and Hilbert spaces; and,
finally, further topics such as functional analysis, distributions and
elements of probability theory.