Unit information: Complex Function Theory in 2020/21

Unit name	Complex Function Theory
Unit code	MATH33000
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. Grava
Open unit status	Not open
Pre-requisites	MATH20006 Metric Spaces
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

To impart an understanding of Complex Function Theory, and facility in its application.

Unit Description

Complex function theory is a remarkably beautiful piece of pure mathematics, and at the same time an indispensable tool in number theory and in many fields of applied mathematics and mathematical methods.

Of central interest are mappings of the complex plane into itself which are differentiable. The property of differentiability alone is enough to guarantee that the function can be represented locally in a power series, in stark contrast to the real-variable theory. This shows that complex analysis is in some ways simpler than real analysis.

The integration theory for complex differentiable functions is highly geometric in nature. Moreover, it provides powerful tools for evaluating real integrals and series. The logarithm and square-root functions on the complex plane are multiple-valued; we shall briefly indicate how they can be seen as single-valued when considered to live on the associated Riemann surface.

The theory of conformal transformations is of great importance in the geometrical theory of differential equations, and has interesting applications in potential theory and fluid dynamics; we shall outline the beginnings of these.

Relation to Other Units

This unit aims for rigorous development and extension of material which has been introduced in the complex function theory part of Multivariable Calculus and Methods of Complex Functions.

Intended Learning Outcomes

At the end of the unit students should:

• be able to recall all definitions and main results,
• be able to give an outline proof of all results,
• be able to give detailed proofs of less involved results,
• be able to apply the theory in standard situations,
• be able to use the ideas of the unit in unseen situations.

Transferable skills:

Problem solving and logical analysis.

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Recommended

• John B. Conway, Functions of One Complex Variable, Springer, 1978
• Jerrold E. Marsden, Basic Complex Analysis, W. H. Freeman, 1999
• Ian Stewart and David Tall, Complex Analysis, Cambridge University Press, 1983

Further

• Murray R. Spiegel, Schaum's Outlines: Complex Variables, McGraw Hill, 2009