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Up: Year 3
Previous: MC382 Abstract Algebra
MC383 Complex Analysis
| Credits: 20 |
Convenor: Dr J. Hunton |
Semester: 2 |
| Prerequisites: |
essential: MC146, MC240, MC248 |
|
| Assessment: |
Regular coursework: 10% |
Three hour exam: 90% |
| Lectures: |
36 |
Classes: |
10 |
| Tutorials: |
none |
Private Study: |
104 |
| Labs: |
none |
Seminars: |
none |
| Project: |
none |
Other: |
none |
| Total: |
150 |
|
|
Explanation of Pre-requisites
The student will be assumed to be familiar with the
general notion of continuity of a real function as
well as other basic concepts from real analysis, such as
differentiability of real functions, power series and integration.
Course Description
In many ways, the subject of Complex Analysis is
aesthetically more pleasing than Real Analysis,
several of the results being ``cleaner'' than their
real counterparts.
In this course, we begin with the study of analogues
for complex functions of familiar properties of
real functions, though differences in the two theories
emerge as we proceed.
Cauchy's theory of complex integration
is developed, culminating in a number of remarkable results and
strikingly beautiful applications.
Towards the end of the course,
the results from complex integration theory are used
to evaluate certain real integrals and to
sum certain real infinite series.
Aims
To help the student to develop an appreciation of the rigorous development
of this remarkable subject, and an understanding of the fundamental
results of the subject.
Objectives
To enable the student to :
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.
Transferable Skills
This course should assist the student in developing skills
of rigorous and precise mathematical writing and expression.
Syllabus
Review of complex numbers.
Basic topological concepts. Complex functions of a real variable.
Complex power series and some functions defined by them.
Radius of converegence, term-by-term differentiability of functions
defined by a power series with positive radius of convergence.
Cauchy-Riemann equations.
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.
Reading list
Essential:
Background:
Recommended:
I. Stewart and D. Tall,
Complex Analyis,
Cambridge University Press.
H.A. Priestley,
Introduction to Complex Analysis,
Oxford University Press.
Details of Assessment
The final assessment of this module will consist of 10% coursework
and 90% from a three hour examination during the Summer exam
period. The 10% coursework contribution will be determined by students'
solutions to coursework problems. The examination paper will contain
8 questions with full marks on the paper obtainable from 5 complete answers.
Next: Year 4
Up: Year 3
Previous: MC382 Abstract Algebra
S. J. Ambler
11/20/1999