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## ECM3703 - Complex Analysis (2015)

MODULE TITLE | Complex Analysis | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM3703 | MODULE CONVENER | Dr Gihan Marasingha (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 11 | 0 | 0 |

Number of Students Taking Module (anticipated) | 29 |
---|

Complex analysis is one of the most beautiful and complete theories in mathematics. Invented in the 19th century by Gauss, Cauchy and Riemann, it has developed into a powerful tool, indispensable to all mathematicians, pure and applied. The skill of computing integrals by means of residue calculus is a major tool in integration and it is an invaluable tool in physics and engineering. In this module, you will see the theory developed in a logical way, emphasizing the wide range of useful applications. You will learn to develop complex analysis in a logical and satisfying way that provides insight into the geometric and topological foundations of the subject. This skill is useful to engineers as it helps you to understand that taking a different viewpoint may render a difficult problem easy.

Prerequisite module: ECM2701 or equivalent

The main aim of this module is to inspire a genuine engagement with complex analysis and its applications in other branches of mathematics

On successful completion of this module, **you should be able to**:

**Module Specific Skills and Knowledge:**

1 understand basic topological concepts like connectedness;

2 comprehend the theory of holomorphic functions, their power-series and integrals;

3 compute contour integrals and to apply this to real analysis.

**Discipline Specific Skills and Knowledge:**

4 compute integrals by means of residue calculus which is a major tool in integration and an invaluable tool in physics, engineering etc;

5 recognise a couple of useful techniques for the computation of integrals with complex methods.

**Personal and Key Transferable/ Employment Skills and Knowledge:**

6 appreciate that taking a different viewpoint may render a difficult problem easy;

7 value the use of theoretical knowledge in concrete problems.

- geometry of the complex plane;

- review of functions of a complex variable: polynomials, rational functions, elementary transcendental functions;

- open and closed sets in the complex plane, continuous curves, path connectedness, domains;

- regular functions on a domain;

- continuity, differentiability, Cauchy-Riemann equations;

- complex power series;

- radius of convergence;

- properties of power series, including differentiation within circle of convergence;

- piecewise continuously differentiable curves;

- contour integrals;

- primitives;

- existence of primitives for simply connected domains;

- Cauchy's Theorem;

- Cauchy integral formulae;

- Taylor series and Laurent series;

- isolated singularities;

- poles, removable and essential singularities;

- maximum modulus principle, Liouville's theorem, fundamental theorem of algebra, meromorphic functions, residue theorem;

- Rouche's theorem, principle of the argument;

- applications to definite integrals, summation of series and the location of zeros.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
---|

Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures/example classes |

Guided independent study | 117 | Working on set mathematical problems |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Not applicable | |||

Coursework | 20 | Written Exams | 80 | Practical Exams |
---|

Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|---|

Written exam – closed book | 80 | 2 hours | All | Specific comments by markers and general comments on website |

Coursework – example sheets | 20 | 60 hours | All | Specific comments by markers and general comments on website |

Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-reassessment |
---|---|---|---|

All above | Written exam (100%) | All | August Ref/Def period |

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Howie, John M | Complex Analysis | Springer | 2003 | 000-1-852-33733-8 | [Library] | |

Set | Priestley H.A. | Introduction to Complex Analysis | Oxford University Press | 2003 | 000-0-198-53428-0 | [Library] | |

Set | Spiegel M.R. | Schaum's outline of theory and problems of complex variables : with an introduction to conformal mapping and its appreciation. | McGraw-Hill | 1981 | 000-0-070-84382-1 | [Library] | |

Set | Stewart I. & Tall D. | Complex Analysis (the hitchhiker's guide to the plane) | Cambridge University Press | 1983 | 000-0-521-28763-4 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2701 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Friday 09 January 2015 | LAST REVISION DATE | Friday 09 January 2015 |

KEY WORDS SEARCH | Complex analysis; contour integration; residue calculus. |
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