# Pacific Math Club Lecture Series: Complex Analysis

Lecturer, *Pacific Math Club*, Fall 2023

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

- 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

- Regular usage of Complex Function Explorer and cplot for plotting functions
- Regular usage of MATLAB and Numpy/Matplotlib for numerical computations

## Topics Covered

- Naive definition of complex numbers
- Euler’s formula \(e^{it} = \cos t+i\sin t\)
- Complex number algebra
- Complex transcendental functions
- Branch cuts and multivalued functions
- Graphing Techniques
- Stereographic projection
- Limits and continuity of complex functions
- Derivatives of complex functions
- Cauchy-Riemann equations
- Line integrals and Green’s theorem
- Line integrals of complex functions
- Cauchy’s theorem and Cauchy’s integral formulae
- Liouville’s theorem and the fundamental theorem of algebra
- Sequences and series of complex numbers
- Taylor and Laurent decompositions
- Classifications of singularities
- Calculus of residues