![]() | Department of Mathematics & Computer Science | |||
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Credits: 20 | Convenor: Dr J. Hunton | Semester: 2 |
Prerequisites: | essential: MC146, MC240, MC248 | |
Assessment: | Regular coursework: 10% | Three hour exam: 90% |
Lectures: | 36 | Problem Classes: | 10 |
Tutorials: | none | Private Study: | 104 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Surgeries: | none | Total: | 150 |
Determine whether a complex function is differentiable.
Define and evaluate contour integrals.
Prove and use Cauchy's integral theorem and Cauchy's integral formula.
Prove the Fundamental Theorem of Algebra.
Prove Taylor's theorem and appreciate that a function which is differentiable in a neighbourhood of a point has a Taylor series expansion about that point.
Prove Laurent's theorem and appreciate that a function which is differentiable in a punctured neighbourhood of a point has a Laurent expansion about that point.
Determine whether a function has a singularity at a given point, and the nature and order of singularities.
Calculate residues and use the residue theorem to evaluate integrals around simple closed paths.
Use the residue theorem to evaluate certain real integrals, to evaluate the sum of certain real series and to assist in certain partial fraction decompositions.
Contour integrals. Simple closed paths. Jordan contours. Star-shaped domains. Cauchy's integral theorem. Cauchy's integral formula. Taylor's theorem. Entire functions. Liouville's theorem and its application to the fundamental theorem of algebra. Laurent series. Singularities, poles, residues. The order of a function at a singularity. Orders of poles. The residue theorem and some techniques for calculating residues. Rouché's theorem. Use of the residue theorem to evaluate real integrals and to evaluate sums of certain real series. Partial fractions.
I. Stewart and D. Tall, Complex Analyis, Cambridge University Press.
H.A. Priestley, Introduction to Complex Analysis, Oxford University Press.
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Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 10/4/2000
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This document has been approved by the Head of Department.
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