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2016　Advanced topics in Analysis C

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Shiga Hiroshige 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Fri3-4(H137)  
Group
-
Course number
MTH.C403
Credits
1
Academic year
2016
Offered quarter
3Q
Syllabus updated
2016/12/14
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

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".

The 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.

Student learning outcomes

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.

Keywords

Elliptic function, Elliptic integral. Riemann surface.

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Riemann's mapping theorem and Schwarz-Christoffer transformation Details will be provided during each class session.
Class 2 Elliptic integral Details will be provided during each class session.
Class 3 Various elliptic integrals Details will be provided during each class session.
Class 4 Elliptic functions and their fundamental properties Details will be provided during each class session.
Class 5 The Jacobi elliptic functions Details will be provided during each class session.
Class 6 Fundamental properties of the Jacobi elliptic functions Details will be provided during each class session.
Class 7 The Weierstrass elliptic function Details will be provided during each class session.
Class 8 Fundamental properties of the Weierstrass elliptic functions Details will be provided during each class session.

Textbook(s)

None.

Reference books, course materials, etc.

To be determined.

Assessment criteria and methods

Repots (100%).

Related courses

  • MTH.C301 ： Complex Analysis I
  • MTH.C302 ： Complex Analysis II
  • MTH.C404 ： Advanced topics in Analysis D

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed MTH.C301 ： Complex Analysis I and MTH.C302 ： Complex Analysis II.

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