module specification

This module extends students’ knowledge of calculus real and complex variable, providing greater depth and rigour for these essential topics.

The aim of the module is to develop a rigorous approach to the analysis of functions of a real and a complex variable and to provide a firm foundation of necessary notions for the student to further their mathematical study..  Further topics in complex integration are also covered and students are introduced to applications of these topics to improper integration of functions of a real variable
Pre-requisite knowledge: MA5011 Further Calculus and MA5052 Differential Equations (studied or Co-requisite)

MA5011 Further Calculus and MA5052 Differential Equations (studied or Co-requisite)

Functions of Real Variable.
Limits, continuity and differentiability of functions of a real variable. Mean Value Theorem, Taylor’s theorem with Lagrange remainder. Taylor and Maclaurin series of standard functions.
Functions of Complex Variable
Limits, continuity and differentiability of functions of a complex variable. Exponential, trigonometric, hyperbolic and logarithmic functions. Cauchy Riemann equation. Contour integration.  Applications to integrals of real variables and to improper integrals.

The module will be delivered through a weekly 3-hour block consisting of a mixture of a lecture and a tutorial. Students’ blended learning is directed via face-to-face learning activities centred on lectures and seminars. There is full provision of documents related to the module in electronic format that can be accessed by students on the WebLearn. The documents include lecture notes, guidance to further reading, exercises and tests.
Teaching methods include a range of tutor led seminar discussions, student led discussions, small group discussions and exercises and lectures given by tutor. The teaching methods will support the main aim of encouraging independent lifelong learning. Tutorials will be student centred using carefully graduated exercises to build up student's confidence and self-esteem and will provide the opportunity for students to reinforce learning and demonstrate their skills and receive individual advice from their tutor.
The tutorial sessions and workshops will also provide an ideal setting for students to meet up with their group members on regular basis and carry out their discussions and investigations..

On successful completion of the module, students should be able to
LO1   Relate the intuitive definitions of a limit, of continuity and of differentiability to more
          formal rigorous definitions of these concepts.  Determine whether given functions
          are continuous or differentiable over given interval.
LO2   To use the mean value Theorem and Cauchy’s Theorem to investigate intervals
          where a function increases (decreases) and to evaluate various limits using
          L’Hopital rules.
LO3   Apply Cauchy Riemann equations to decide whether a given function is
          differentiable.  Evaluate contour integrals using parametrisation and apply the
          residue theorem to evaluate contour integrals.
LO4   To apply contour integrals to improper integration of functions of a real variable

Core Text:
1. R. Haggarty (1989). Fundamentals of Mathematical Analysis. Addison-Wesley
2. R. Churchill and J. Brown (1990). Complex Variables and Applications.  McGraw Hill
Recommended Reading:
3. R. Adams (1999). A complete Course in Calculus. Addison-Wesley.
4. J. Stewart (1998). Calculus; Concepts and Contexts. Brooks/Cole.
5. T. Osborne (1999), Complex Variables and Their Applications. Addison-Wesley
6. N. Alexandrou, Lecture Notes on Complex Variable, London Metropolitan University