MMath&Phys Mathematics and Physics

On successful completion of the course, students will be able to:

• Demonstrate properties of holomorphic (regular) functions, such as the Cauchy-Riemann equations, and apply the Cauchy-Riemann Theorem to elementary functions

   

• Define, analyse and make use of elementary functions including those defined by power series
• Use the Cauchy-Taylor Theorem and Laurent's Theorem to expand a holomorphic function as a power series on a disc or on an annulus
• Identify the location and nature of a singularity of a complex function and, in the case of a pole, to calculate the order and the residue, and justify properties of the residue

• Define and calculate complex integrals using a variety of methods, such as the Fundamental Theorem of Countour Integration and the Cauchy Residue Theorem, and apply to the evaluation of some real integrals.

   

This course introduces the calculus of complex functions of a complex variable. Complex differentiability is a very strong condition and differentiable functions have many strong properties. Integration is along paths in the complex plane. The central result of this spectacularly beautiful part of mathematics is Cauchy’s Theorem guaranteeing that certain integrals along closed paths are zero. This striking result leads to useful techniques for evaluating real integrals based on the calculus of residues.

Syllabus:

1. Complex functions: Domains and paths in the complex plane. Differentiation and the Cauchy-Riemann equations; holomorphic functions. Path integrals and the Fundamental Theorem of contour integration.  

2. Power series: Power series and their radius of convergence. Derivatives and integrals of power series. Elementary functions such as exp, sin, cos, sinh, cosh and log.

3. Cauchy’s Theorem and Formula: Winding numbers, Cauchy’s theorem, Cauchy’s integral formula and the estimation lemma.

4. Taylor and Laurent series: The Cauchy-Taylor theorem, Liouville’s theorem, Laurent’s theorem and calculation of Laurent series.

5. Residues: Isolated singularities, poles and their residues; Cauchy’s residue theorem. Applications of the residue theorem to the evaluation of trigonometric integrals, integrals over the real line, and summation of series.

Online coursework test (using stack)
Test mid-way through semester; Stack feedback available after test is closed
20% Weighting

Written exam
General feedback provided after exam is marked
80% Weighting

Stewart, Ian. (1983). Complex analysis: the hitchhiker's guide to the plane. Cambridge University Press. ISBN: 9781139171632

Lang, Serge. (1998). Complex analysis. Springer. ISBN: 0387985921

Copson, E. T. (1935). An introduction to the theory of functions of a complex variable. Clarendon. ISBN: 0198531451

Ahlfors, Lars V. (1979). Complex analysis: an introduction to the theory of analytic functions of one complex variable. McGraw-Hill. ISBN: 0070006571 

Tuition fees for home students commencing their studies in September 2027 will be £10,050 per annum.

For general information please see the undergraduate finance pages.

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.

For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .