Module Code - Title:

MS4045 - COMPLEX ANALYSIS

Year Last Offered:

2025/6

Hours Per Week:

Lecture

2

Lab

0

Tutorial

1

Other

0

Private

7

Credits

6

Grading Type:

N

Prerequisite Modules:

MS4022

Rationale and Purpose of the Module:

To introduce the concept of an analytic function of a complex variable and integration on the complex plane.

Syllabus:

Single- and multi-valued functions, branch points and branch cuts; analytic functions, the Cauchy-Riemann equations; Laurent series, poles and essential singularities; Cauchy's Integral Theorem, Cauchy's Integral Formula; the Residue Theorem, the Estimation Lemma, Jordan's Lemma, integration of functions with branch points; conformal mappings; analytic continuation.

Learning Outcomes:

Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

1. Demonstrate proficiency with complex functions including multi-valued functions and branch cuts. (Mid-term and end of semester assessment.) 2. Demonstrate understanding of analytic functions and their properties; the Cauchy-Riemann equations and the Cauchy Integral Theorem. (Mid-term and end of semester assessment.) 3. Demonstrate ability to integrate single- and multi-valued functions on the complex plane, including those arising from the inverse Laplace transformation, using the Estimation Lemma and Jordan's Lemma. (Mid-term and end of semester assessment.) 4. Demonstrate understanding of conformal mappings and ability to use them to map circles, polygons, etc. into a complex half-plane.

Affective (Attitudes and Values)

None

Psychomotor (Physical Skills)

None

How the Module will be Taught and what will be the Learning Experiences of the Students:

Lectures and tutorials

Research Findings Incorporated in to the Syllabus (If Relevant):

Prime Texts:

Zill, D. G. and Shanahan, P. D. (2003) A first course in complex analysis , Jones and Bartlett Publishers

Ablowitz, M. J. and Fokas, A. S. (2011) Complex Variables: Introduction and Applications (2nd ed.) , Cambridge University Press

Kreyszig, E. (2011) Advanced engineering mathematics , Wiley

Other Relevant Texts:

Beck, M., Marchesi, G., Pixton, D., Sabalka, L. (2002) A first course in complex analysis , Orthogonal Publishing

Programme(s) in which this Module is Offered:

BSMSCIUFA - MATHEMATICAL SCIENCES

BSFIMAUFA - FINANCIAL MATHEMATICS

BSMAPHUFA - MATHEMATICS AND PHYSICS

Semester(s) Module is Offered:

Autumn

Module Leader:

Eugene.Benilov@ul.ie