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Taken directly from Canvas: Taken directly from Canvas:
# Exam topics # 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. 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 C and R^2. - Similarities and differences between $\mathbb{C}$ and $\mathbb{R}^2$.
- Holomorphic functions and their properties. - Holomorphic functions and their properties.
- Complex exponential and logarithm. - Complex exponential and logarithm.
- Integration in the complex plane. - Integration in the complex plane.
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- Singularities and their classification. - Singularities and their classification.
- Residues and the residue theorem. - Residues and the residue theorem.
- Applications to the computation of integrals. - Applications to the computation of integrals.
# Relevant Questions # Relevant Questions
**Here are 20 relevant exam questions in the topology and measure theory part of the course:** Here are 20 relevant exam questions in the topology and measure theory part of the course:
What distinguishes the real numbers from the rational ones? 1. What distinguishes the real numbers from the rational ones?
2. What is an equivalence relation?
What is an equivalence relation? 3. What is a topological space? Examples?
4. What is the ball topology on a metric space?
What is a topological space? Examples? 5. What is the topology on a Banach space?
6. What is a compact set?
What is the ball topology on a metric space? 7. State the Heine-Borel theorem. Proof?
8. What is a continuous function?
What is the topology on a Banach space? 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.
What is a compact set? 11. What is the initial topology?
12. What is the product topology?
State the Heine-Borel theorem. Proof? 13. What is a measure? Easy examples?
14. Define the Lebesgue integral of a extended non-negative measurable function.
What is a continuous function? 15. State Lebesgue's monotone convergence theorem.
16. Define $L^p$-spaces, and point out their crucial property.
Why does a real valued continuous function obtain its maximum on a compact set? 17. (Not relevant: State the Riesz representation theorem.
18. What is the Lebesgue measure on $\mathbb{R}^n$ ?
What is a net? Given an example of an upward filtered ordered set. 19. What is a complex measure?
20. State the Lebesgue-Radon-Nikodym theorem.)
What is the initial topology?
What is the product topology?
What is a measure? Easy examples?
Define the Lebesgue integral of a extended non-negative measurable function.
State Lebesgue's monotone convergence theorem.
Define L^p-spaces, and point out their crucial property.
(Not relevant: State the Riesz representation theorem.
What is the Lebesgue measure on R^n ?
What is a complex measure?
State the Lebesgue-Radon-Nikodym theorem.)