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Complex Function Theory 34 (MATH M3000)

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Administrative Information

  1. Unit number and title: MATH M3000 Complex Function Theory 34
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2006-07
  6. Unit Organiser: Yuri Netrusov
  7. Lecturer: Yuri Netrusov
  8. Teaching block: 1
  9. Prerequisites: MATH 20200 Metric Spaces.

Unit aims

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

General Description of the Unit

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 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.

Teaching Methods

  • 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.

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

Assessment Methods

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 are NOT permitted.

Award of Credit Points

Credit points are gained by:

  • either passing the unit (i.e. gaining a mark of 50 or over),
  • or getting an examination mark of 30 or over, and also handing in satisfactory attempts at the project and five specified homework questions, to be given out at various times over the semester.

Transferable Skills

Logical analysis and problem solving

Texts

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

  1. I. Stewart and D. Tall, Complex Analysis , Cambridge University Press
  2. J. E. Marsden, Basic Complex Analysis , W. H. Freeman
  3. S. Lang, Complex Analysis , Springer
  4. 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.

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.

The Project

  • The project involves independent reading from books as well as solving advanced exercises. The project is supervised by the lecturer of the course. The student agrees a project topic with the lecturer at the beginning of the first term. Possible topics
  • Analytical proof of the Cauchy Theorem (following Dixon).
  • Homological version of the Cauchy Theorem (following Artin).
  • Maximum principles. Hadamard and Phragmen-Lindelöf Theorems.
  • Subharmonic and superharmonic functions. The Dirichlet problem on general domains.
  • The range of analytic functions. The Great Picard Theorem.
  • Mapping properties of analytic functions. The Riemann Mapping Theorem.
  • Analytic continuation and elementary Riemann surfaces.
  • Other topics may also be discussed with the lecturer. The second half of the book [4] can be a good source.
  • The length of the written report might be about 5000 words.
  • Written report on the project should be submitted during the 1st week of the 2nd term. During the 2nd week of the 2nd term the student make a presentation of the project on a seminar, which is attended by all students taking Comlex Function Theory 34.