Registry

Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Complex Analysis and Applications
Module Code	MS206
School	School of Mathematics
Online Module Resources
Module Co-ordinator	Semester 1: Michael Clancy (maths) Semester 2: Michael Clancy (maths) Autumn: Michael Clancy (maths)
Module Teacher	Michael Clancy (maths)
NFQ level	8	Credit Rating	5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

This module introduces students to the theory of complex analysis, and enables the students to use the powerful methods of complex analysis to solve problems arising in applications. Students are presented with the basic properties of complex numbers and then the main theorems of complex analysis are derived in a fashion that enables students to understand the proofs of the results and to apply the results competently to solve problems which arise in applications. The module provides students with a platform of knowledge that allows them to engage in further study of material which requires complex analysis to be a prerequisite, and gives a good grounding in rigorously proving mathematical theorems and then using the results to solve problems.Students will participate in the following learning activities:Lectures:Students will attend a series of lectures.Problem-solving: Students will attend weekly tutorials to engage in problem-solving exercises.Reading: Students are expected to use the textbooks mentioned below.

Learning Outcomes

1. Define basic properties of the complex numbers and solve exercises which test a knowledge of the properties.
2. Define elementary functions of complex numbers and compute relations involving elementary functions and their inverses.
3. Compute derivatives of complex functions and perform the integration of complex functions over contours in the plane.
4. Construct and interpret various power series of complex functions.
5. Prove a selection of important theorems and results for complex functions, particularly results concerning differentiation and integration.
6. Apply these theorems and results to solve a variety of problems arising in applications.

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecture	24	Weekly lecture
Tutorial	11	Supervised work on exercises
Independent learning time	90	Recommended time for self-study
Total Workload: 125

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Review of complex numbers.
Algebraic and geometrical representation of complex numbers. Euler's formula, rational powers, inequalities..

Functions of a complex variable..
Domains, regions, interior points, closure. Continuity and differentiability of functions, the Cauchy-Riemann equations. Analytic and harmonic functions. The definition and properties of elementary functions: exp function, log function, trigonometric and hyperbolic functions, and their inverses. Definition and properties of complex exponents..

Integration of complex functions.
Integration in the complex plane, bounds on integrals. Cauchy's theorem. Cauchy's integral formula, Cauchy's inequality. The maximum modulus principle. Schwarz's lemma..

Power series of complex numbers.
Convergence of sequences and series. Taylor series and Laurent series of a complex function. Absolute and uniform convergence of power series. Integration and differentiation of power series..

Theory of residues..
Evaluation of real integrals by contour integration round poles. Zeros and poles of complex funcions. Application to the gamma function and inversion of Laplace transform..

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Short answer questions	Class tests	25%	n/a

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List

J.W. Brown and R. V.Churchill: 0, Complex Variables and Applications, McGrawHill,
J. E. Marsden and M. H. Hoffman: 0, Basic Complex Analysis, Greeman,
S. Lang: 0, Complex Analysis, Springer,

Other Resources

None

Array

Programme or List of Programmes

ACM

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BSSA

Study Abroad (DCU Business School)

BSSAO

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CAFM

Common Entry into Mathematical Sciences