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MATH30040

Academic Year 2025/2026

Complex Analysis (MATH30040)

Subject:
Mathematics
College:
Science
School:
Mathematics & Statistics
Level:
3 (Degree)
Credits:
5
Module Coordinator:
Professor Maria Meehan
Trimester:
Spring
Mode of Delivery:
On Campus
Internship Module:
No
How will I be graded?
Letter grades

Curricular information is subject to change.

Complex numbers were introduced by mathematicians working in the 16th century. Complex analysis, which first emerged in the 19th century largely due to the pioneering work of Augustin-Louis Cauchy, is a branch of mathematical analysis that focuses on differentiable (more precisely 'analytic' or 'holomorphic') complex-valued functions defined on the set of complex numbers and certain subsets of it. As well as being a rich and beautiful theory in its own right, complex analysis has innumerable applications in physics, engineering and other areas of mathematics such as analytic number theory. We will cover topics chosen from the following list:

1. Differentiation on the complex plane; exponential, trigonometric and logarithmic functions;
2. Path integration; Cauchy's Theorem and Cauchy's Integral Formula;
3. Power series; Taylor's Theorem and its consequences;
4. Laurent series; classification of singularities; Cauchy's Residue Theorem, and
5. Applications of the Residue Theorem, such as Fourier Transforms and the evaluation of real integrals and infinite sums.

About this Module

Learning Outcomes:

On successful completion of this module, the student is expected to be able to state the definitions, explain the proofs of the theorems and generate examples of the definitions and theorems covering the topics listed above, and carry out associated computations. These include, but are not limited to, stating the main theorems in the module, together with some of their consequences; prove a selection of these results; apply the Cauchy-Riemann equations and Cauchy's Integral Formula; evaluate the integral of a complex function along a path; compute Taylor and Laurent expansions of various functions; compute residues and contour integrals; and evaluate Fourier Transforms and real integrals and infinite sums using residues.

Student Effort Hours:
Student Effort Type Hours
Lectures

26
Tutorial

10
Specified Learning Activities

36
Autonomous Student Learning

28
Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.

Requirements, Exclusions and Recommendations
Learning Requirements:

Prior knowledge of complex numbers, differentiation of functions of one variable, partial derivatives, line/path integration, continuity of functions of one variable, and infinite series will be assumed.

In particular, the student should have taken a level one module on calculus of one variable and a level two module on calculus of several variables.


Module Requisites and Incompatibles
Incompatibles:
MST30050 - Complex Analysis


 

Assessment Strategy
Description Timing Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Quizzes/Short Exercises: Class Test Week 8 Standard conversion grade scale 40% No
20
 No
Exam (In-person): 2 hour End of Trimester exam End of trimester
Duration:
2 hr(s)
 Standard conversion grade scale 40% No
80
 No

Carry forward of passed components
No
 

Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 

Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.
Spring Tutorial Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Fri 09:00 - 09:50
Spring Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Fri 12:00 - 13:50
Spring Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Mon 11:00 - 11:50
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