Pacific Math Club Lecture Series: Complex Analysis

Ran a weekly lecture series through University of the Pacific’s mathematics club to lecture on topics from a typical complex analysis course which is not offered at University of the Pacific. Topics were presented in a way friendly to those with background in calculus at the levels of MATH 051, MATH 053, and MATH 055 at the University of the Pacific.

Resources used

1. Handouts written by myself that are inspired by or reference the following books:
   • Complex Variables and Applications (9th edition) by Brown and Churchill
   • Visual Complex Analysis by Needham
   • Complex Analysis by Gamelin
   • Visual Complex Functions by Wegert
   • Complex Analysis by Stein and Shakarchi
2. Regular usage of Complex Function Explorer and cplot for plotting functions
3. Regular usage of MATLAB and Numpy/Matplotlib for numerical computations

Topics Covered

1. Naive definition of complex numbers
2. Euler’s formula \(e^{it} = \cos ⁡t+i\sin ⁡t\)
3. Complex number algebra
4. Complex transcendental functions
5. Branch cuts and multivalued functions
6. Graphing Techniques
7. Stereographic projection
8. Limits and continuity of complex functions
9. Derivatives of complex functions
10. Cauchy-Riemann equations
11. Line integrals and Green’s theorem
12. Line integrals of complex functions
13. Cauchy’s theorem and Cauchy’s integral formulae
14. Liouville’s theorem and the fundamental theorem of algebra
15. Sequences and series of complex numbers
16. Taylor and Laurent decompositions
17. Classifications of singularities
18. Calculus of residues