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MMath&Phys Mathematics and Physics / Course details
Year of entry: 2024
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Course unit details:
2P1: Complex Analysis
|Available as a free choice unit?
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’.
|Foundations of Pure Mathematics A
|Calculus and Vectors A
|Real Analysis A
The unit aims to introduce the basic ideas of complex analysis, with particular emphasis on Cauchy’s Theorem and the calculus of residues
On succesful completion of the course, students will be able to:
- Prove the Cauchy-Reimann Theorem and its converse and use them decide whether a given function is holomorphic
- Use power series to define a holomorphic function and calculate its radius of convergence, and perform computations with such series. Define elementary holomorphic functions such as sin. cos. sinh. cosh. exp. log.
- Define the complex integral and use a variety of methods (the Fundamental Theorem and the Cauchy Residue Theorem) to calculate the complex integral of a given function
- Use Taylor's Theorem and Laurent's Theorem to expand a holomorphic function in terms of power series on a disc and Laurent series on an annulus respectively
- Identify the location and nature of a singularity of a function and, in case of poles. Calculate the order and the residue
- Apply techniques from complex analysis to deduce results in other areas of mathematics. Including proving the Fundamental Theorem of Algebra and calculating infinite real integrals, trigonometric integrals and the summation of series
Teaching and learning methods
2 hours of lectures and 1 hour tutorial per week, for weeks 1-11 of semester 2.
In-class coursework test
Test mid-way through semester; marked scripts returned within 15 days
General feedback provided after exam is marked
Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983.
|Scheduled activity hours
|Independent study hours
This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.
Please see Blackboard / course unit related emails for any further updates