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## MATH2090 Real and Complex Analysis

### 10 creditsClass Size: 120

Module manager: Dr Alison Parker
Email: parker@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2011/12

### Pre-requisite qualifications

MATH1035 or (MATH1050 and MATH1055), or equivalent.

### This module is mutually exclusive with

 MATH2015 Analysis 2

This module is approved as an Elective

### Module summary

Complex analysis was the great triumph of nineteenth century mathematics. The results of the French mathematician Cauchy laid the foundations for many deep results and applications to other branches of mathematics. The latter part of this course is an exposition of Cauchy's beautiful and surprising theorems about analytic functions. One such result enables us to use integration in the complex plane to calculate definite integrals which apparently do not involve the complex numbers at all! The first part of the course does some necessary spadework, deepening and extending ideas and results about continuity and differentiability of real-valued functions.

### Objectives

- To deepen the understanding of ideas based on limits.
- To introduce the basic ideas of complex analysis.
- To show that many ideas of analysis, such as convergence of series, have their most natural setting in the complex plane, and to illustrate the application of these ideas to problems in real analysis.

On completion of this module, students should be able to:
(a) make simple arguments concerning limits of real-valued functions; show continuity and differentiability in real-valued functions; and make simple uses of these;
(b) calculate Taylor and Laurent expansions and use the calculus of residues to evaluate integrals.

### Syllabus

1. Real Analysis Improper integrals (infinite range only); limits, continuity and differentiability of functions of a real variable.
2. Basic ideas of complex function theory. Limits, continuity, analytic functions, Cauchy-Riemann equations.
3. Contour integrals. Cauchy's theorem, Cauchy's integral formula.
4. Power series. Analytic functions represented as Taylor or Laurent series. Singularities. Orders of poles, Cauchy's residue theorem, evaluation of definite integrals.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 22.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated