Course description
The second half of a two semester Masters level class on Complex Analysis. We will take material from the textbook of Marsden and Hoffman starting at Chapter 5. This includes conformal mapping and series and product expansions. Afterwards, there will be topics as time permits, hopefully including parts of Fourier analyis that use complex variable theory and some stuff on elliptic integrals and genus 1 Riemann surfaces (tori). Aside from the specific material, a goal is to gain facility in matheamtical analysis and to be exposed to ideas that are developed in active parts of mathematics including topology, number theory and partial differential equations.
- Review
- Contours and integrals
- Connected and simply connected domains
- Cauchy formula, local regularity, Liouville, etc.
- Conformal mapping
- Riemann mapping theorem
- Schwarz Christoffel formula
- Applications to harmonic functions
- Infinite series and products
- Partial fractions, Mittag Leffler, Euler/Bernoulli sums
- Weierstrass P function
- Product representations, Hadamard, Euler/Bernoulli (again)
- Euler products for gamma and zeta functions
- Fourier analysis
- Fourier and Taylor/Laurent series
- Fourier transform, delta function, regularization
- Poisson summation formula, Theta inversion formula, etc.
- Stationary phase and saddle point approach to oscillatory integrals
- Elliptic functions
- Elliptic integrals and genus 1 Riemann surfaces, periods
- Properties of doubly periodic functions
Prerequisites:
A first course in Complex variables equivalent to our Complex Variables I.
Books:
- (main text) Basic Complex Analysis, Marsden and Hoffman
- (highly recommended) Complex Analysis, Alfors
- (highly recommended) Complex Analysis, Stein and Shakarchi
- (historical interest) A Course of Modern Analysis, Whittaker and Watson
- (deeper analysis) Real and Complex Analysis, Rudin
- (traditional textbook) Complex Variables and Applications, Brown and Churchill
Assignments, exams, grading:
The final grade will be based on weekly homework assignments and an in-class written final exam.
Communication:
Please use the Brightspace site for content and homework communications. This way everyone sees and benefits from questions and answers, and there can be class discussion. related comm. Email the instructor for issues that do not involve others such as scheduling appointments, homework extensions, advice, etc.
Academic integrity:
Students are encouraged to explore and collaborate widely to understand the material. This includes looking at print and online sources and interacting with experts and each other. Students may receive some help with assignments, but each student must create (write up, code, run) solutions individually. Students may not share ("borrow" or lend) assignment solutions -- all writing must be done individually. Students may not plagairize solutions from other sources such as books or web sites. Assignments may not be crowd-sourced on any web platform. Do not post homework exercises. Violation of these policies may result in grade lowering or more serious penalties, depending on severity.