This is an archived handbook for a previous year. The current undergraduate and postgraduate handbooks can be viewed using the links above.
Undergraduate Handbook 2010-2011 (archived)
Module MATH4151: ELLIPTIC FUNCTIONS IV
Department: Mathematical Sciences
MATH4151: ELLIPTIC FUNCTIONS IV
| Type | Level | Credits | Availability | Module Cap | Location |
|---|---|---|---|---|---|
| Open | 4 | 20 | Available in 2010/2011 | None. | Durham |
Pre-requisites
- Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3 and including Complex Analysis II (MATH2011).
Corequisites
- None.
Excluded Combinations
- Elliptic Functions III (MATH3221).
Aims
- To introduce the theory of multiply-periodic functions of one complex variable and to develop and apply it.
Content
- Elliptic integrals, sums of squares, simple pendulum.
- Elliptic functions: Review of necessary parts of complex analysis.
- Doubly periodic functions.
- Liouville's theorems.
- Weierstrass elliptic functions: Differential equation.
- Laurent expansion.
- Addition theorems, elliptic function identities.
- Abel-Jacobi theorem.
- Theta functions.
- Elementary theory of infinite products.
- Poisson summation formula.
- Jacobi triple produce identity, Jacobi derivative formula.
- Relation with Weierstrass elliptic functions.
- Modular forms and functions Definition of the space of modular forms (level 1 holomorphic forms only.
- Finite-dimensionality, formula for the dimension.
- (NB the syllabus is identical to ELLIPTIC FUNCTIONS III (A) which is taught in parallel).
Learning Outcomes
Subject-specific Knowledge
- By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Elliptic Functions.
- have an understanding of specialised and complex theoretical mathematics in the field of Elliptic Functions.
- have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: General properties of elliptic functions.
- Weierstrass elliptic functions.
- Theta functions.
- Modular functions.
Subject-specific Skills
- In addition students will have highly specialised and advanced mathematical skills in the following areas: Spatial awareness, abstract reasoning.
Key Skills
Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module
- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
- Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired and the ability to solve complex and specialised problems.
Teaching Methods and Learning Hours
| Activity | Number | Frequency | Duration | Total Hours | Attendance Monitored |
|---|---|---|---|---|---|
| Lectures | 40 | 2 per week | 1 Hour | 40 | |
| Preparation and Reading | 160 | ||||
| Total | 200 |
Summative Assessment
| Component | Component Weighting | Element | Length/Duration | Element Weighting | Resit Opportunity |
|---|---|---|---|---|---|
| Examination | 100% | three-hour examination | 100% |
Formative Assessment
Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.
Students who do not attend monitored activities shown under Teaching Methods
and Learning Hours, or who fail to complete the summative or formative
assessment(s) specified above, may be subject to the Academic Progress
procedures defined in the University's General Regulation V, and may be
required to leave the University.