
Module Availability 
COMPLEX VARS submodule: Autumn Semester REAL ANALYSIS 2 submodule: Spring Semester 


Assessment Pattern 
COMPLEX VARS submodule: Assignment 1: 3.5 % (7% of submodule mark) Test: 5% (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) Test: 5% (10% of submodule mark) Assignment 2: 5% (10% of submodule mark) Written examination (2 hours, unseen): 37.5 % Qualifying conditions: 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.



Module Overview 
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. 


Prerequisites/Corequisites 
CALCULUS MAT1015 (short name MS114)) PROOF, PROBABILITY AND EXPERIMENT MAT1017 (short name: MS125) REAL ANALYSIS I MAT1003 (short name: MS107)



Module Aims 
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.



Learning Outcomes 
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 CauchyRiemann 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 Calculus, etc.). * Argue logically to justify proofs or give counterexamples of properties of continuity, convergence, differentiability and integrability. * Calculate Taylor series and determine and justify its convergence and convergence of some other sequences of functions (e.g. power series).



Module Content 
Complex Analysis submodule: Complex functions, complex differentiability; CauchyRiemann 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 to contour 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, inverse functions. * 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 Riemann integral. 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, power 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 the convener.
3 contact hours per week for 10 weeks for both, the Complex Analysis and the Real Analysis submodules.



Selected Texts/Journals 
For the Complex Analysis submodule: Recommended:  I. Stewart and D. Tall, Complex analysis, CUP, (1987).  M.R. Spiegel, Complex Variables, Schaum's outline, McGraw Hill, (1981).
Further Reading:  J. Marsden and M. Hoffman, Basic Complex Analysis, Freeman & co (1998).
For the Real Analysis submodule: Recommended:  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)
Other texts:  J. Lewin, Mathematical Analysis, Cambridge University Press (2003).  P.E. Kopp, Analysis, Arnold Publishers, (1990).  S. Lang, Analysis I, AddisonWesley (1968).



Last Updated 
16/6/2008


