Autumn semester

Components of Assessment	Method(s)	Percentage Weighting
Coursework	Made up of one mid-term test and one assignment	25 %
Examination	Written examination (2 hours, unseen).	75 %

MS104 Techniques in Calculus 2

At the end of the course 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.

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.

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.

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).

15 August 2006