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
      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).

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
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).

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.

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, Addison-Wesley (1968).