Module Content

This module will present theorems and proofs to rigorously back up ideas related to functions and calculus and will extend concepts in real analysis to complex numbers. This will include: 

Differentiation: definition of a derivative; derivatives of sums, products, and compositions of functions; Rolle’s Theorem; the mean value theorems; L’Hôpital’s rule; inverse functions; higher derivatives; Taylor’s theorem. 

Integration: the Riemann integral; classes of integrable functions; properties of integrals; the mean value theorem; the fundamental theorem of calculus; techniques of integration; improper integrals of the first and second kind. 

Particular functions: the logarithmic and exponential functions, circular functions 

(sine and cosine) 

Introduction to Complex Analysis: What complex numbers are; properties of complex numbers; complex functions and continuity; differentiability of complex functions.

Teaching Methods

The module will be delivered on campus, with weekly lecture and tutorial sessions. 

Printed notes will be given ahead of time for each section of the course, to support and enhance students’ preparation and engagement during class sessions. Lectures will follow the notes, with discussions of the main theoretical topics, and study of examples of the applications of the theory. There will be a strong emphasis on student involvement in discussions in lectures, to encourage a more active approach to learning the material, and to allow the delivery to be tailored to build on the students’ current understanding. 

Regular formative work in tutorial sessions will allow students to internalise the mathematical ideas and methods developed in the lectures, and lead to the development of problem-solving skills. This formative work will also feed back into the delivery of lectures and tutorials.

Assessment Methods

This module is assessed through a portfolio of exercises and an examination.