Complex Variables (Jan 2022 - May 2022)

This is a one semester course on complex variables for MAC students cohort I. The reference used in the course is "COMPLEX VARIABLES AND APPLICATIONS, 8th edition" by James Ward Brown and Ruel V. Churchill. This course is intended to cover at least from chapter 1 to chapter 5.

The following is the link to join the class: https://join.skype.com/JuVdOw1lLKaS.

The How to Prove is an example showing how to prove things in mathematics. This serves as an idea or a method to do proofs in mathematics.

The homework have to be sent to my email (found it in About) with only one pdf file under the name "ComplexV_HW1" and so on. You are expected to work together and teach each other to solve these problems.

Class on 16-Jan-2022: Covered chapter 1

Homework 1: (Group Homework, Deadline 30-Jan-2022 at 00:00 Phnom Penh time)

1. Exercise 8 in section 3.

2. Exercise 15 in section 5.

3. Exercise 11 in section 8.

4. Exercise 6 in section 11.

5. Exercise 7 in section 11.

6. Prove that a set $S$ is closed if and only if $S$ contains each of its accumulation points if and only if the closure of $S$ is $S$ if and only if the complement of $S$ ( that is $\mathbb{C}\setminus S$) is open.

7. Prove that the sets $(b)$ and $(d)$ in exercise 1 of section 11 are domains and the set in exercise 5 of section 11 is not a domain. Find the closures of thses domains.

8. Exercise 10 in section 11.

Class on 30-Jan-2022: Covered chapter 2: Sections 12-16

Homework 2: (Group Homework, Deadline 13-Feb-2022 at 00:00 Phnom Penh time)

1. Give an example of a boundary point that is not an accumulation point. This exercise shows that the set of accumulation points of a set $S$ is not the union of $S$ with its boundary (this corrects an error in the class). In fact, $S'\subset S\cup bd(S)$.

2. Prove that $z$ is in the closure of a set $S$ if and only if each neighbourhood of $z$ contains at least one point in $S$. This exercise shows the difference between the closure point and the accumulation point. We can also conclude that any accumulation point is a closure point.

3. Prove that the closure of a set is the smallest closed set containing that set. This exercise gives another equivalent definition of the closure of a set.

4. A complex sequence $z_n$ coverges to $z$ if $\forall \epsilon>0,\exists N\in\mathbb{N}$ such that $|z_n -z|< \epsilon$ whenever $n \geq N$. Prove that if $z_0 \in S\subset \mathbb{C}$ is an accumulation point then there exists a sequence in $S$ that converges to $z_0$. This is the reason that in some book, an accumulation point is called a limit point.

5. Prove that $\lim\limits_{z\to z_0}f(z)=w_0$ if and only if $\forall\epsilon>0,\forall N\in \mathbb{N},\exists\delta>0$ such that $|f(z)-w_0|< N\epsilon$ whenever $0<|z-z_0|<\delta$.

6. Prove that $\lim\limits_{z\to z_0}f(z)=w_0$ if and only if $\forall\epsilon>0,\forall N\in \mathbb{N},\exists\delta>0$ such that $|f(z)-w_0|< \frac{\epsilon}{N}$ whenever $0<|z-z_0|<\delta$.

7. Exercise 4 in section 14.

8. Exercise 7 in section 14.

9. Exercise 1 on page 55.

10. Exercise 2 on page 55.

11. Exercise 3 on page 55.

12. Exercise 5 on page 55.

13. Exercise 7 on page 56.

14. Exercise 8 on page 56.

15. Exercise 9 on page 56.

Class on 13-Feb-2022: Covered chapter 2: sections 17-25

Homework 3: (Group Homework, Deadline 27-Feb-2022 at 00:00 Phnom Penh time)

1. Exercise 10 on page 56.

2. Exercise 13 on page 56.

3. Exercise 4 on page 62.

4. Exercise 6 on page 72.

5. Exercise 7 on page 72.

6. Exercise 8 on page 72.

7. Exercise 4 on page 77.

8. Exercise 5 on page 77.

9. Exercise 6 on page 78.

10. Exercise 7 on page 78.

Class on 27-Feb-2022: Covered chapter 3

Homework 4: (Group Homework, Deadline 13-Mar-2022 at 00:00 Phnom Penh time)

1. Exercise 3 on page 92.

2. Exercise 11 on page 92.

3. Exercise 14 on page 92.

4. Exercise 6 on page 97.

5. Exercise 8 on page 97.

6. Exercise 9 on page 97.

7. Exercise 1 on page 100.

8. Exercise 2 on page 100.

9. Exercise 6 on page 100.

10. Exercise 8 on page 104.

11. Exercise 9 on page 104.

12. Exercise 4 on page 108.

13. Exercise 6 on page 108.

14. Exercise 8 and 9 on page 112.

15. Exercise 4 on page 115.

Class on 13-Mar-2022: Covered chapter 4: Sections 37-39

Homework 5: (Group Homework, Deadline 27-Mar-2022 at 00:00 Phnom Penh time)

Do all exercises in sections 38 and 39.

Class on 27-Mar-2022: Covered chapter 4: Sections 40-42

Homework 6: (Group Homework, Deadline 10-Apr-2022 at 00:00 Phnom Penh time)

Do all exercises in section 42.

Class on 24-Apr-2022: Covered chapter 4: Sections 43-54

Homework 7: (Group Homework, Deadline 12-May-2022 at 00:00 Phnom Penh time)

1. Exercise 3 in section 43.

2. Exercise 4 in section 43.

3. Exercise 6 in section 43.

4. Exercise 1 in section 45.

5. Exercise 5 in section 45.

6. Exercise 2 in section 49.

7. Exercise 4 in section 49.

8. Exercise 6 in section 49.

9. Exercise 1 in section 52.

10. Exercise 2 in section 52.

11. Exercise 7 in section 52.

12. Exercise 1 in section 54.

13. Exercise 3 in section 54.

14. Exercise 6 in section 54.

15. Exercise 8 in section 54.

Class on 15-May-2022: Read chapter 5

Homework 8: (Group Homework, Deadline 05-Jun-2022 at 00:00 Phnom Penh time)

1. Section 56: Exercise 4, Section 59: Exercise 13, Section 62: Exercise 8, Section 66: Exercise 12, Section 67: Exercise 8.

2. Section 56: Exercise 6, Section 59: Exercise 10, Section 62: Exercise 5, Section 66: Exercise 10, Section 67: Exercise 7.

3. Section 56: Exercise 9, Section 59: Exercise 14, Section 62: Exercise 11, Section 66: Exercise 5, Section 67: Exercise 6.

4. Section 59: Exercise 6, Section 62: Exercises 2 and 9, Section 66: Exercise 2, Section 67: Exercise 5.

5. Section 59: Exercise 7, Section 62: Exercises 4 and 10, Section 66: Exercise 1, Section 67: Exercise 3.