Course Introduction

• Prerequisite： calculus, basic mathematics, advanced calculus
• Recommended for： junior
• Introduction：

Complex Analysis is an important area with many application in Physics, Number Theory, Combinatorics, etc. The main goal of complex analysis is to study analytic functions. In this undergraduate course, student will learn the basics of the theory. Topics covered include analytic functions, line integrals, singularities, the residue theorem, conformal mappings, analytic continuation, etc.

• Syllabus：

1. Definition of Complex Numbers.
   
2. Topology of the Complex Plane.
   
3. Power Series.
   
4. Analytic Functions and Cauchy-Riemann Equations.
   
5. Line Integrals.
   
6. Closed Curve Theorem for Entire Functions.
   
7. Properties of Entire Functions.
   
8. Properties of Analytic Functions
   
9. Cauchy’s Theorem
   
10. Isolated Singularities.
    
11. Laurent Series.
    
12. The Residue Theorem.
    
13. Introduction into Conformal Mappings.
    
14. Riemann Mapping Theorem.
    
15. Analytic Continuation.

• Reference：

1. J. Bak and D. J. Newman (1999). Complex Analysis, Springer, corr. 2nd printing edition.
   
2. H. A. Priestley (2003). Introduction to Complex Analysis, OUP Oxford 2nd edition.