Undergraduate Programme and Module Handbook 2008-2009 (archived)

# Module MATH3221: ELLIPTIC FUNCTIONS III

## Department: Mathematical Sciences

### MATH3221: ELLIPTIC FUNCTIONS III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2008/09 and alternate years thereafter | Module Cap | None. | Location | Durham |
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#### Prerequisites

- (Complex Analysis II (MATH2011) and one extra 20 credit Level 2 mathematics module) OR (Complex Analysis II (MATH2011) and Core Mathematics B1 (if taken in Year 2)).

#### Corequisites

- One 20 credit Level 3 mathematics module.

#### Excluded Combination of Modules

- Elliptic Functions IV (MATH4151).

#### 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 product 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.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve novel and/or complex problems in Elliptic Functions.
- have a systematic and coherent understanding of theoretical mathematics in the field of Elliptic Functions.
- have acquired 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 forms.

Subject-specific Skills:

- In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: 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 unpredictable problems of some complexity.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 40 | 2 per week for 19 weeks and 2 in term 3 | 1 Hour | 40 | |

Preparation and Reading | 160 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

three hour written 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.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University