This is the second part of the
basic graduate sequences course in analysis. You can find the lecture
notes for the first part here.
course we will study the behavior of differentiable complex-valued
functions f(z) of a complex variable z. The key idea in
the course is that complex differentiability is a much more restrictive
condition than real differentiability. In fact,
complex-differentiable functions are so rigid that the entire behavior of
such a function is completely determined if you know its values even on a
tiny open set. One understands these rigidity properties by making
use of contour integration - integration along a path in the complex
theory gains its force because there are plenty of interesting functions to
which it applies. All the usual functions - polynomials, rational
functions, exponential, trigonometric functions, and so on - are
differentiable in the complex sense. Very often, complex analysis
provides the solution to "real variable" problems involving these
functions; as someone said, "The shortest path between two real points
often passes through the complex domain." Moreover, complex analysis
is a key tool for understanding other "higher transcendental
functions" such as the Gamma function, the Zeta function, and the
elliptic functions, which are important in number theory and many other
parts of mathematics. A secondary aim of this course is to introduce
you to some of these functions.
One of the surprises of complex
analysis is the role that topology plays. Simple
questions like "do I choose the positive or negative sign with the
square root" turn out to have surprisingly subtle answers, rooted in
the notion of the fundamental group of a topological space (which you will
be looking at in the Topology and Geometry course parallel to this). These
topological notions eventually culminate in the notion of a Riemann
surface as the correct global context for complex analysis. We
will not develop this idea fully, but we will discuss `multiple-valued
functions' and their branch points; again, we will try to illustrate how
these exotic-sounding concepts help in doing practical calculations.
We will be using ANGEL (Penn State's course management software)
for this course and you will need to log on to ANGEL regularly throughout
the semester. You can connect to the ANGEL main page by
pressing the button below. You will find the course syllabus and
other relevant information on the ANGEL page for the course.
Some course items will be linked
directly from this page; they will all be available through ANGEL also.
Lecture notes (full set) reformatted for easier printing.