Course Catalogue

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The aim of this course is to introduce students to complex functions and their applications.

Lectures at 1.00 pm on Fridays some Thursdays. Tutorials fortnightly, at a time to be arranged. 

Maths 2A and 2D at Grade D3 or above.
Please note: this is one of a package of 4 level-3 courses in Mathematics leading to a designated degree in Mathematics.
Full details of the requirements for a designated degree can be found in the University Calendar.
The requirements for the designated degree include a second-year curriculum that includes Mathematics 2A, 2B, 2D and another level 2 Mathematics course. An average grade of D3 over these 4 level-2 courses is required.

Methods of Complex Analysis (85JN)

90% Examination, 10% Coursework.

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below. 

The aim of this course is to introduce students to complex functions and their applications. It will focus on properties of analytic functions, contour integration and conformal maps.

By the end of this course students will be able to:

■ state the Cauchy-Riemann equations; use the Cauchy-Riemann equations to determine whether a complex function is analytic on a specified domain; determine whether a function is harmonic and calculate a harmonic conjugate; define and compute with elementary (polynomial, rational, exponential, trigonometric) complex functions; define and compute complex logarithms and powers;

■ compute integrals of continuous functions along curves in the complex plane;

■ state Cauchy's theorem for a simple closed path, Taylor's theorem and Cauchy's integral formula for the n-th derivative;

■ determine the nature of singularities and compute the residues at poles of suitable meromorphic functions; state the residue theorem and use the residue theorem to evaluate real integrals using methods from the course;

■ define the notion of a conformal map; give examples of conformal maps between elementary simple domains; establish properties of bilinear maps; compute the image of a line or circle under a bilinear transformation.

Students will be deemed to have completed the course (gaining 20 credits and at least a Grade H) if they have: a 70% or better record of attendance in tutorials; handed in written work when set; taken the degree (or resit) exam. Students who have not satisfied the above requirements may still be deemed to have completed the course subject to approval by the Head of School. Explanations are required for absences of more than a few days. Where relevant, medical certificates should be produced.