Methods of Complex Functions
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.
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.
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.
Linking abstract to visual / geometric explanations, problem solving, assimilation of abstract ideas and application to particular problems.
- Functions of one complex variable. Holomorphic functions. Cauchy-Riemann condition.
- Integral Calculus. Cauchy's theorems of integration. Liouville's theorem.
- Power series. Taylor's Theorem. Laurent's Theorem.
- Residues. Isolated singularities. Residue Theorem.
- 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
Linear Algebra and Geometry, Analysis 1A, Analysis 1B and Calculus 1.
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.
Further exam information can be found on the Maths Intranet.