MAT 514

Syllabus and Schedule

The textbook for this course is A First Course in Complex Analysis, by Matthias Beck, Gerald Marchesi, Dennis Pixton and Lucas Sabalka. (I will refer to this book as [BMPS]).
Another reference that can be useful is Basic Complex Analysis, by Jerrold E. Marsden and Michael J. Hoffman (I will refer to this book as [MF]). There are many worked examples in this book, so it is worth looking at.

Syllabus

1. Functions on the Plane.
1.1 Complex Numbers and Topology of the Plane.
1.2 Differentiable Functions in the Plane.
1.3 Holomorphic Functions in the Plane.

2. Integration and Cauchy's Theorems.
2.1 Review of Integration.
2.2 Cauchy's Theorem.
2.3 Applications of Cauchy's Theorem.
2.4 Power series.

3. The Residue Theorem.
3.1 Laurent series.
3.2 Calculation of Residues.
3.3 The Residue Theorem.
3.4 Applications of the Residue Theorem.

Schedule

Below there is the intended schedule for the course. It can change depending on how we progress on the contents (the goal of this course is to be useful to you, not to go fast to cover as many topics as possible).

Tuesday July 11: Presentation of the course. Complex Numbers. Topology of the Plane.
Suggested reading: Chapter 1 in [BMPS], 1.1 and 1.2 in [MF].

Thursday July 13: Topology of the Plane. Review of Limits and differentiability. Holomorphicity. Examples.
Suggested reading: 1.4, 2.1 and 2.2 in [BMPS], 1.3 and 1.4 in [MF].

Tuesday July 18: Holomorphicity and basic properties. Examples of Holomorphic functions.
Suggested reading: 3.3, 3.4 and 3.5 in [BMPS], 1.5 and 1.6 in [MF].

Thursday July 20: Holomorphicity and basic properties. Examples of Holomorphic functions.
Suggested reading: 3.1, 3.2 in [BMPS], 1.5 and 1.6 in [MF].

Tuesday July 25: Review of Integration and basic properties. Cauchy's theorem.
Suggested reading: 4.1 and 4.2 in [BMPS], 2.1 in [MF].

Thursday July 27: Presentation day. Cauchy's theorem(s) and integration.
Homework: (to be posted).
Suggested reading: 4.3, 4.4, 5.1 in [BMPS], 2.2 and 2.4 in [MF].

Tuesday August 1: Power series. Examples.
Suggested reading: 7.1, 7.2, 7.3 and 7.4 in [BMPS], 3.2 in [MF].

Thursday August 3: Power series and Holomorphic functions. Examples.
Suggested reading: 8.1, 8.2 and 8.3 in [BMPS], 3.2 in [MF].

Tuesday August 8: The Residue theorem and applications.
Suggested reading: 9.1 and 9.2 in [BMPS], 4.1 and 4.2 in [MF].

Thursday August 10: Applications of the Residue theorem.
Suggested reading: 10.1 in [BMPS], 4.3 and 4.4 in [MF].

Tuesday August 15: Presentation day and Review session (it will be on Zoom).
Homework: (to be posted).

Thursday August 17: Final exam.
Practice problems: (to be posted).

Information for students with disabilities
If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support Services at (631) 632-6748 or http://studentaffairs.stonybrook.edu/dss/. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the following website: http://www.stonybrook.edu/ehs/fire/disabilities.shtml