This course is designed for advanced undergraduate students and first year graduate students who have had first course in complex analysis, and can be used as all or part of a second course in complex variables. In our exposition we have also kept in mind the potential reader interested in self-study, someone in the physical sciences or technology with a reasonable degree of proficiency and experience in mathematics, or even a researcher in pure mathematics. This course will be followed by "Advanced topics in Analysis D".
Our subject matter, primarily, is the Jacobi elliptic functions and the Weierstrass elliptic functions and their interrelation with Riemann surfaces. Our purpose for basing a treatment of elliptic functions on Riemann surface theory is twofold. On the one hand, elliptic functions are indissolubly wedded to elliptic integrals, and for an intelligent discussion of the latter in the complex domain the use of Riemann surfaces is really essential. On the other hand, for the student who wants to learn a little about Riemann surface theory, either for its applications to other areas or for itself, the connection with elliptic functions, particularly the Jacobi functions, forms a very natural and concrete path of introduction.
By the end of this course, students will be able to:
1) understand elliptic integrals,
2) understand fundamental properties of elliptic functions.
3) understand Jacobi and Weierstrass elliptic functions.
Elliptic function, Elliptic integral. Riemann surface.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | The same as [MTH.C403 : Advanced topics in Analysis C]. | Details will be provided during each class session. |
Class 2 | Elliptic integral | |
Class 3 | Various elliptic integrals | |
Class 4 | Elliptic functions and their fundamental properties | |
Class 5 | The Jacobi elliptic functions | |
Class 6 | Fundamental properties of the Jacobi elliptic functions | |
Class 7 | The Weierstrass elliptic function | |
Class 8 | Fundamental properties of the Weierstrass elliptic functions |
None.
To be determined.
Reports (100%).
Students are expected to have passed ZUA.C301 : Complex Analysis I and ZUA.C302 : Exercises in Analysis B I