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.

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

Relevant Questions

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?
2. What is an equivalence relation?
3. What is a topological space? Examples?
4. What is the ball topology on a metric space?
5. What is the topology on a Banach space?
6. What is a compact set?
7. State the Heine-Borel theorem. Proof?
8. What is a continuous function?
9. Why does a real valued continuous function obtain its maximum on a compact set?
10. What is a net? Given an example of an upward filtered ordered set.
11. What is the initial topology?
12. What is the product topology?
13. What is a measure? Easy examples?
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.
17. (Not relevant: 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.)