Complex Analysis I

Summary module description:
This module provides an introduction to complex analysis.

Aims:
To introduce students to complex analysis and enable them to use complex variable techniques, particularly in some cases where the original problem does not involve complex numbers.

Assessable learning outcomes:
By the end of the module students are expected to be able to:
- solve problems involving holomorphic functions
- recognise and be able to apply the complex exponential and logarithm
- evaluate path integrals of complex functions
- identify singularities and residues of holomorphic functions
- calculate appropriate real integrals using complex techniques

Additional outcomes:
By the end of the module the student will begin to understand and recognise some of the structure of holomorphic functions.

Outline content:
Differentiable functions of a complex variable are remarkably well-behaved, and most of the technical complications of the real case do not arise with complex functions. This leads to some remarkably powerful results, and it turns out that complex variable techniques often offer the simplest method of evaluating certain real integrals. The notion of complex differentiability relates closely with power series. Contour integration in the complex plane will be introduced and the remarkable theorem of Cauchy established, from which a whole range of applications follow. Some applications to the evaluation of real integrals are given.

Brief description of teaching and learning methods:
Lectures supported by problem sheets and lecture-based tutorials.

Summative Assessment Methods:
Written exam 90%
Set exercise 10%

Other information on summative assessment:
One assignment and one examination paper

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late, in accordance with the University policy.
where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
2 hours.

Requirements for a pass:
A mark of 40% overall.