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2005/06 Undergraduate Module Catalogue

MATH2090 Real and Complex Analysis

10 creditsClass Size: 200

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

Pre-requisites

MATH1031 or MATH1050, or equivalent. Not with MATH2011, MATH2021, MATH2490

This module is approved as an Elective

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

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. 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (22 hours) and examples classes (11 hours).

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (85%), coursework (15%).

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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