# Complex Function Theory 3

## 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.

## Learning objectives

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.

## Syllabus

**Differentiation and integration of complex functions:**Cauchy-Riemann equations, contour integrals, the fundamental theorem of contour integration - a quick survey.**Cauchy's theorems:**Cauchy's theorem for a triangle, Cauchy's theorem for a starshaped domain; homotopy, simply connected domains, Deformation theorem (without proof), Cauchy's theorem for simply connected domains.**Cauchy's integral formula:**Cauchy's formula, Morera's and Liouville's Theorem, fundamental theorem of algebra.**Local properties of analytic functions:**Taylor series, Laurent series.**Zeros and singularities of analytic functions:**classification of zeros and isolated singularities, Casorati-Weierstrass's theorem, behaviour of analytic functions at infinity.**The residue theorem:**the topological index, the residue theorem, Rouche's and the local mapping theorem.**Global properties of analytic functions:**the identity theorem, maximum modulus theorems.**Harmonic functions:**harmonic functions and harmonic conjugates, the Poisson formula, the Dirichlet problem.**Conformal mappings:**basic properties of conformal mappings, the Riemann mapping's theorem (without proof), fractional linear transformations and other standard transformations, application of conformal mappings to Laplace's equation.

## Reading and References

Many books dealing with complex analysis may be found in section QA331 of the Queen's Library. The books:

- I. Stewart and D. Tall,
*Complex Analysis*, Cambridge University Press - J. E. Marsden,
*Basic Complex Analysis*, W. H. Freeman - J. B. Conway,
*Functions of one complex variable*, Springer

may be found particularly useful. The bulk of the course will follow [1] quite closely. The Schaum Outline Series *Complex Variables* by M. R. Spiegel is a good additional source of problems.

**Unit code:** MATH33000

**Level of study:** H/6

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturers:** Dr Witold Sadowski and Nick Jones

## Pre-requisites

MATH20006 Metric Spaces.

## Co-requisites

None

## Methods of teaching

Lecture course, with exercise sheets to be done by students.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in January

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.