Unit information: Complex Function Theory (34) in 2013/14

Unit name	Complex Function Theory (34)
Unit code	MATHM3000
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. van den Berg
Open unit status	Not open
Pre-requisites	MATH20900
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Complex analysis, or the calculus of complex-valued functions, is one of the most beautiful self-contained areas of mathematics. In many ways simpler than real one-variable calculus, it is possible to derive far-reaching results having important scientific applications as well as providing powerful tools in other branches of mathematics. Starting from the idea of differentiability of complex- valued functions through the idea of conformal mappings, leading up to Cauchy's theorem on the integration of complex functions, it proves possible to tackle successfully such diverse problems as two-dimensional potential flows of an ideal fluid or to evaluate explicitly improper real integrals or infinite series.

Aims

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

Syllabus

Lectures:

• 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.

Relation to Other Units

This unit aims for rigorous justification, development and extension of material which has been introduced in the complex function theory part of Calculus 2. Students should have a good knowledge of first year analysis and second year calculus courses.

The unit is based on the same lectures as Complex Function Theory 3, but with additional material. It is therefore not available to students who have taken, or are taking, Complex Function Theory 3.

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,
• have developed their ability to learn new mathematics without lectures, and present this material in writing and as a talk.

Transferable Skills:

Logical analysis and problem solving

Teaching Information

• Lecture course of 30 lectures, with weekly exercise sheets to be done by students. This part of the course is shared with 3rd year students taking CFT3.
• Project on an advanced topic of Complex Function Theory.

Assessment Information

The final assessment mark for Complex Function Theory 34 is calculated as follows:

• 20% from the CFT34 project, which is assessed by a written report (80% of the project mark) and a short talk (20% of the project mark).

• 80% from a 2½-hour examination in April consisting of FIVE questions (the same paper as for Complex Function Theory 3). A candidate's FOUR best answers will be used for assessment. Calculators of an approved type are allowed.

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
• S. Lang, Complex Analysis , Springer
• 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.