|COMPLEX VARS submodule: Autumn Semester
REAL ANALYSIS 2 submodule: Spring Semester
|COMPLEX VARS submodule:
Assignment 1: 3.5 %
(7% of submodule mark)
(10% of submodule mark)
Assignment 2: 4 %
(8% of submodule mark)
Written examination (2 hours, unseen) : 37.5%
REAL ANALYSIS 2 submodule:
Assignment 1: 2.5%
(5% of submodule mark)
(10% of submodule mark)
Assignment 2: 5%
(10% of submodule mark)
Written examination (2 hours, unseen): 37.5 %
An overall aggregate mark of 40% for the module (calculated using the
weights above) AND an aggregate mark of 25% for each submodule, are
required to pass the module.
|This module builds on the level 1 module Real Analysis 1.
It introduces students to the theory of
the functions of a complex variable and
focuses on continuity, differentiability and integrability
of real functions of one variable.
|CALCULUS MAT1015 (short name MS114))
PROOF, PROBABILITY AND EXPERIMENT MAT1017 (short name: MS125)
REAL ANALYSIS I MAT1003 (short name: MS107)
Complex analysis, traditionally known as the theory of functions of
a complex variable, is one of the fundaments of modern mathematics and used
in many branches of mathematics and physics. In the first half of this
module we introduce the main results of this theory.
The objective of the second half of the module is to provide an
introduction to analysis, which is the branch of mathematics that
rigorously studies functions, continuity and limit processes, such as
differentiation and integration. The second half of this module
intends to lead to a deeper understanding of what it means when a
sequence or series is said to converge, or a function to be
continuous. Historic motivation and the rigorous use of definitions
and logic play a central role. Furthermore, tools such as convergence
tests are presented and their validity proved.
At the end of the Complex Analysis submodule the student
should have a thorough understanding of the theory of complex
functions of a complex variable and should be able to apply this
knowledge in a variety of contexts. In particular the student should
be able to:
* Understand what complex differentiation is.
* Quote, derive and apply the Cauchy-Riemann equations.
* Study convergence properties of a complex power series
* Perform contour integrations of continuous functions
* Understand and apply Cauchy's theorem and Cauchy's formula
* Derive analyticity and Liouville's theorem from Cauchy's formula
* apply Taylor's and Laurent's theorems to obtain power series expansions
* Identify and classify singularities of complex functions and
find residues of simple poles.
* Derive and apply the residue theorem to compute real integrals
using residue calculus.
At completion of the Real Analysis sub module, a
student should be able to
* Prove continuity, differentiability and integrability of
function by using the formal definitions and basic properties.
* Quote, prove and apply main theorems in Real Analysis (e.g.,
Intermediate, Extreme and Mean Value Theorems, Rolle's Theorem,
l'Hôpital's rule, Taylor's Theorem, Fundamental Theorem of
* Argue logically to justify proofs or give counterexamples of
properties of continuity, convergence, differentiability and
* Calculate Taylor series and determine and justify its
convergence and convergence of some other sequences of functions
(e.g. power series).
Complex Analysis submodule:
Complex functions, complex differentiability; Cauchy-Riemann equations
and harmonic functions; conformal mappings;
complex integration: derivation of Cauchy's theorem and Cauchy
integration formula, derivation of Liouville theorem and fundamental
theorem of algebra;
application of Taylor expansion and Laurent expansion;
classification of singularities; proof of residue theorem; application
integration and evaluation of real integrals.
The Real Analysis submodule contains the following topics:
* Limits of functions, continuity (formal definition). Sums, products,
compositions. Intermediate value theorem and extreme value theorem.
* Differentiable functions (sums, products,
quotients). Differentiability implies continuity. Chain rule,
* Rolle's theorem, mean value theorem, l'Hôpital's rule. Higher
derivatives. Taylor 's theorem.
* Theory of integration: upper and lower sums and integrals, the
Conditions for integrability (e.g., continuity implies integrability).
Indefinite integration, and the fundamental theorem of
calculus. Taylor series with integral remainder.
* Convergence of sequences of functions (e.g., Taylor series,
|Methods of Teaching/Learning
Teaching is by lectures and tutorials. Learning takes place through lectures,
tutorials, exercises and background reading. Summary notes are provided by
3 contact hours per week for 10 weeks for both, the Complex Analysis and
the Real Analysis submodules.
For the Complex Analysis submodule:
- I. Stewart and D. Tall, Complex analysis, CUP, (1987).
- M.R. Spiegel, Complex Variables, Schaum's outline, McGraw Hill, (1981).
- J. Marsden and M. Hoffman, Basic Complex Analysis, Freeman & co (1998).
For the Real Analysis submodule:
- J. M. Howie, Real Analysis, Springer (2001)
(Available in paperback from UniS bookshop, Springer or amazon.co.uk)
- J.E. Snow and K.E. Weller, Exploratory Examples for Real Analysis, Cambridge University Press (2004).
(Available in paperback from UniS bookshop, CUP or amazon.co.uk)
- J. Lewin, Mathematical Analysis, Cambridge University Press (2003).
- P.E. Kopp, Analysis, Arnold Publishers, (1990).
- S. Lang, Analysis I, Addison-Wesley (1968).