Methods of Complex Functions

Unit aims

The unit gives an overview over methods for differentiating and integrating complex-­valued functions, introduces the underlying theoretical results, and shows how they can be applied to problems in complex and real analysis.

Unit description

The unit introduces functions of a complex variable, with a focus on holomorphic functions. It extends elementary calculus to functions of a complex variable, showing similarities and differences between the properties of two-­dimensional vector fields and functions of a complex variable. The emphasis is on basic ideas and methods; theorems will be stated rigorously and the theory will be carefully developed, but the emphasis is on methods rather than proofs.

Relation to other units

This unit feeds into pure and applied mathematics, such as Complex Function Theory (which develops the material) and Fluid Dynamics. Applied Partial Differential Equations and Mathematical Methods use the material.

Learning objectives

Be familiar with and be able to use the elementary properties of holomorphic functions of a complex variable. Find power series expansions, integrate holomorphic and functions with and without singularities. Master residue calculus, and apply it to real-­valued integrals.

Transferable skills

Linking abstract to visual / geometric explanations, problem solving, assimilation of abstract ideas and application to particular problems.

Syllabus

  1. Functions of one complex variable. Holomorphic functions. Cauchy-Riemann condition.
  2. Integral Calculus. Cauchy's theorems of integration. Liouville's theorem.
  3. Power series. Taylor's Theorem. Laurent's Theorem.
  4. Residues. Isolated singularities. Residue Theorem.
  5. Solving real-valued integrals using the Residue Theorem

Reading and References

Jerrold E. Marsden & Michael J. Hoffman, Basic Complex Analysis, ed. 3 , W. H. Freeman & Company, 1999. Lecture notes will be made available.

Unit code: MATH20001
Level of study: I/5
Credit points: 10
Teaching block (weeks): 1 (7-12)
Lecturer: Dr Karoline Wiesner

Pre-requisites

Linear Algebra and Geometry, Analysis 1A, Analysis 1B and Calculus 1.

Co-requisites

None

Methods of teaching

Lectures, problems classes, homework and solutions (issued later).

Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

  • 100% from a 1 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.

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