Unit name | Complex Function Theory (34) |
---|---|

Unit code | MATHM3000 |

Credit points | 20 |

Level of study | M/7 |

Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |

Unit director | Dr. Netrusov |

Open unit status | Not open |

Pre-requisites |
MATH 20200 Metric Spaces |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Unit aims

To impart an understanding of Complex Function Theory, and facility in its application.

General Description of the Unit

Complex function theory is a remarkably beautiful piece of pure mathematics, and at the same time an indispensable tool in number theory and in many fields of applied mathematics and mathematical methods.

Of central interest are mappings of the complex plane into itself which are differentiable. The property of differentiability alone is enough to guarantee that the function can be represented locally in a power series, in stark contrast to the real-variable theory. This shows that complex analysis is in some ways simpler than real analysis.

The integration theory for complex differentiable functions is highly geometric in nature. Moreover, it provides powerful tools for evaluating real integrals and series. The logarithm and square-root functions on the complex plane are multiple-valued; we shall briefly indicate how they can be seen as single-valued when considered to live on the associated Riemann surface.

The theory of conformal transformations is of great importance in the geometrical theory of differential equations, and has interesting applications in potential theory and fluid dynamics; we shall outline the beginnings of these.

Relation to Other Units

This unit aims for rigorous justification, development and extension of material which has been introduced in the complex function theory part of Calculus 2. Students should have a good knowledge of first year analysis and second year calculus courses.

The unit is based on the same lectures as Complex Function Theory 3, but with additional material. It is therefore not available to students who have taken, or are taking, Complex Function Theory 3.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Learning Objectives

At the end of the unit students should:

- be able to recall all definitions and main results,
- be able to give an outline proof of all results,
- be able to give detailed proofs of less involved results,
- be able to apply the theory in standard situations,
- be able to use the ideas of the unit in unseen situations,

have developed their ability to learn new mathematics without lectures, and present this material in writing and as a talk.

Transferable Skills

Logical analysis and problem solving.

- Lecture course of 30 lectures, with weekly exercise sheets to be done by students. This part of the course is shared with 3rd year students taking CFT3.
- Project on an advanced topic of Complex Function Theory.

80% Examination and 20% Project.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Many books dealing with complex analysis may be found in section QA331 of the Queen's Library. The books:

- I. Stewart and D. Tall, Complex Analysis , Cambridge University Press
- J. E. Marsden, Basic Complex Analysis , W. H. Freeman
- S. Lang, Complex Analysis , Springer
- J. B. Conway, Functions of one complex variable , Springer

may be found particularly useful. The bulk of the course will follow [1] quite closely.

The Schaum Outline Series Complex Variables by M. R. Spiegel is a good additional source of problems.