MA552 (for undergraduate courses only)
OverviewThis module is concerned with complex functions, that is functions which are both defined for and assume complex values. Their theory follows a quite different development from that of real functions, is remarkable in its directness and elegance, and leads to many useful applications.Topics covered will include: Complex numbers. Domains and simple connectivity. Cauchy-Riemann equations. Integration and Cauchy's theorem. Singularities and residues. Applications.
This module appears in:
48 (approx.. 36 lectures and 12 example classes).
Method of assessment
80% Examination, 20% Coursework
M.R. Spiegel Complex Variables, McGraw-Hill, 1964
H.A. Priestley Introduction to Complex Analysis, Oxford University Press, 2003
J.H. Mathews & R.W Howell Complex Analysis for Mathematics and Engineering, Jones and Bartlett 5th ed., 2006
I Stewart & D Tall, Complex Analysis, Cambridge, 2004
The intended subject specific learning outcomes
On successful completion of this module students will:
a) Have a reasonable ability to perform basic computational skills: calculations with Cartesian and polar form of complex numbers, modulus and argument; roots of unity; partial fractions and the general binomial theorem; calculations with exponential, trigonometric and hyperbolic functions, complex logarithm and complex exponents, and hyperbolic functions.
b) Have a reasonable knowledge, and understand the place in the theory and the proofs: of the Cauchy Fundamental Theorem, Cauchy Integral Formulae with and without winding numbers, the Deformation Theorem, Existence and formulae for Taylor and Laurent series, differentiability of power series, Cauchy Residue Theorem, the Cauchy-Riemann equations, a proof of the Fundamental Theorem of Algebra..
c) Gain experience and solve problems using more advanced analytic skills such as: computation of Taylor and Laurent series; radius of convergence of power series; calculation of residues and types of singularity; evaluation of integrals using residues, possibly including the use of Riemann surfaces; homotopy of paths to ease calculations of path integrals; use of winding numbers of paths; evaluation of limits and differentiability of a complex function; conjugate harmonic functions.
The intended generic learning outcomes
Students who successfully complete this module will have further developed:
a) a logical mathematical approach to solving problems;
b) an ability to solve problems relevant to applications in engineering and physics;
c) the basic skills for postgraduate studies in topology, engineering mathematics and applied analysis.