Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
| MATH1000 | Core Mathematics |
| MATH1110 | Real Analysis |
MATH3017 Calculus in the Complex Plane and half of MATH2051 (curves part) of Geometry of Curves and Surfaces
This module is not approved as an Elective
The ideas of real and complex analysis are powerful products of nineteenth- and twentieth-century pure mathematics: interesting and beautiful in their own right and with application to theoretical physics. Analysis is concerned with rigorously describing, understanding and quantifying properties of functions and maps (which for instance model particles, their trajectories, the shape of space and physical laws).
This module will introduce students to real functions of several variables and the calculus of complex functions of one variable. By the end of the module students will have an intuitive grasp and hands-on competency with the rigorous study of curves, differentials, line integrals, complex derivatives and some applications of this toolkit.
On successful completion of the module students will have demonstrated the following learning outcomes:
1. Identify open and closed subsets of R^n.
2. Determine whether functions of several variables are continuous, or not.
3. Describe and interpret derivatives of functions of several variables from a range of perspectives.
4. Use the Cauchy-Riemann equations to decide whether a given function is holomorphic.
5. Compute contour integrals: from first principles; using the fundamental theorem of the calculus; using Cauchy's residue theorem; and using Cauchy's integral formula.
6. Compute the Laurent series of a holomorphic function about an isolated singularity.
7. Classify the singularities of holomorphic functions and compute, in the case of a pole, its order and residue.
8. Evaluate typical definite integrals by using the calculus of residues.
9. Identify regularly parametrized curves.
10. Compute basic properties of curves, such as curvature.
11. Deduce properties of curvature from pictures of curves.
Skills Learning Outcomes
On successful completion of the module students will have demonstrated the following skills learning outcomes:
a. Identify and set academic goals.
b. Use a range of sources to aid learning.
c. Communicate technical information
d. Manage workload and deadlines through prioritisation and productivity skills.
1. Open, closed and complete subsets of R^n.
2. Continuous functions of several variables.
3. Differentiable functions of several variables.
4. The inverse function theorem and the implicit function theorem.
5. Contour integrals and the fundamental theorem of calculus.
6. Holomorphic functions and the Cauchy-Riemann equations.
7. Cauchy's theorem and its consequences.
8. Taylor's theorem and the Laurent series.
9. Residues and the evaluation of integrals.
10. Curvature and arclength of parametrized curves.
11. Signed curvature of planar curves; Frenet equations for space curves.
Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar:
12. Taylor series in several variables.
13. The Newton-Raphson method in several variables.
14. The method of characteristics for partial differential equations.
15. Harmonic conjugate functions.
16. Evolutes and involutes of planar curves.
Methods of assessment
The assessment details for this module will be provided at the start of the academic year
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 22 | 2 | 44 |
| Seminar | 10 | 1 | 10 |
| Private study hours | 146 | ||
| Total Contact hours | 54 | ||
| Total hours (100hr per 10 credits) | 200 | ||
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team