Complex Analysis

COURSE OBJECTIVE
- The student can decide whether a complex function is analytic (=differentiable in the complex sense) and knows the connection with the
Cauchy-Riemann equations.
- She can do computations with elementary functions such as exp/log/sin/cos over the complex numberts.
- She can integrate analytic functions along a path on the complex plane, using the theorem of Cauchy-Goursat and its corollaries.
- She can compute Laurent series and determine the type of singularities of analytic functions.
- She can compute integrals of complex functions using the residue theorem and knows how to use this to compute integrals of real
functions.

COURSE CONTENT
In complex analysis one generalizes the standard concepts of real analysis such as differentiation and integration from the real line to the complex plane. Although these generalizations arise very naturally and all standard examples of functions are also differentiable in the complex sense, the latter property surprisingly turns out to be much stronger. As a consequence, complex differentiable functions immediately obey very special properties which we are going to explore in this course. In particular, they lead to completely new and efficient methods for computing integrals of real functions.

During the lectures the following topics will be treated:
- complex differentiation and the Cauchy-Riemann equations
- complex integration and the theorem of Cauchy-Goursat
- elementary properties of complex differentiable functions
- singularities, Laurent series and the residue theorem
- application to integrals of real functions

TEACHING METHODS
Lecture (2 hours) and tutorial class (2 hours)

TYPE OF ASSESSMENT
Two written exams (40%+40%) and two hand-in homeworks (10%+10%). The retake exam counts for 100% of the final grade.

RECOMMENDED BACKGROUND KNOWLEDGE
Calculus, Analysis, Linear algebra