Undergraduate Programme and Module Handbook 2009-2010 (archived)

# Module MATH4151: ELLIPTIC FUNCTIONS IV

## Department: Mathematical Sciences

### MATH4151: ELLIPTIC FUNCTIONS IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2010/11 and alternate years thereafter | Module Cap | None. | Location | Durham |
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#### Prerequisites

- 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 Combination of Modules

- 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 | |
---|---|---|---|---|---|

Lectures | 40 | 2 per week | 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 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