ACIT4330-Page/content/Exam Preparation.md

Taken directly from Canvas:

Exam topic

The following list is meant to provide a starting point for the type of questions (relating to the complex function theory part of the course) you will get at the exam (but this is not an exhaustive list). You are not expected to know the details of the proofs, but you need to show that you understand the concepts and know how to apply them.

Relevant Questions

Topology and Measure Theory

Here are 20 relevant exam questions in the topology and measure theory part of the course:

1. What distinguishes the real numbers from the rational ones? Revision/Real Analysis/Question 1
2. What is an equivalence relation? Revision/Real Analysis/Question 2
3. What is a topological space? Examples? Revision/Real Analysis/Question 3
4. What is the ball topology on a metric space? Revision/Real Analysis/Question 4
5. What is the topology on a Banach space? Question 5
6. What is a compact set? Question 6
7. State the Heine-Borel theorem. Proof? Question 7
8. What is a continuous function? Question 8
9. Why does a real valued continuous function obtain its maximum on a compact set? Question 9
10. What is a net? Given an example of an upward filtered ordered set. Question 10
11. What is the initial topology? Question 11
12. What is the product topology? Question 12
13. What is a measure? Easy examples? Question 13
14. Define the Lebesgue integral of a extended non-negative measurable function.
15. State Lebesgue's monotone convergence theorem.
16. Define $L^p$-spaces, and point out their crucial property.

Not relevant

17. State the Riesz representation theorem.
18. What is the Lebesgue measure on \mathbb{R}^n ?
19. What is a complex measure?
20. State the Lebesgue-Radon-Nikodym theorem.

Complex Analysis

• Similarities and differences between \mathbb{C} and \mathbb{R}^2. Revision/Complex Analysis/Question 1
• Holomorphic functions and their properties. Revision/Complex Analysis/Question 2
• Complex exponential and logarithm. Revision/Real Analysis/Question 3
• Integration in the complex plane. Revision/Real Analysis/Question 4
• Cauchy's theorem and integral formula. Revision/Real Analysis/Question 5
• Taylor and Laurent series.
• Singularities and their classification.
• Residues and the residue theorem.
• Applications to the computation of integrals.