Mathematics 503 Midterm Oral Exam Information 2017

The oral exam will ask questions about the topics covered so far in
the course, which consists of Chapters I,II,II of the text with the
exception of sections II.5.5 and III.7.  You should understand the
connections between the main theorems and how they are applied to
understand complex functions.
 
Among the topics of interest are

Cauchy Riemann equations
Cauchy integration formula
Examples and constructions of holomorphic functions and harmonic functions
Properties of holomorphic functions on connected open sets
Behavior of functions holomorphic on a punctured disk
Contour integrals, residue theorem and applications
Open mapping theorem, maximal principle
The argument principle and Rouche's theorem
Existence of logarithms of functions

The exam will last 30 minutes and will consist of questions that will test
whether you know the basic theorems, why they are true and how they are used.
Some questions will ask for examples, others will concern consequences of
the basic theorems.

Sample questions:

1. Is there a function holomorphic in a neighborhood of zero that takes
the value 1/n at 1/3n and at 1/(3n+2) for infinitely many n?

2. Show that a function f(z) holomorphic on a disk about 0 which is real on
the intersection of the real line with the disk is given by a power series 
with real coefficients.

3. What is Morera's theorem?  What is the idea of the proof?  

4. Explain how you would compute the integral of  cos(x)/(x^2+1)^2 over the
real line.