This module starts by further developing the theory of continuous and differentiable functions of one real variable introduced in Real Analysis. The basic theory of integration on a closed bounded interval is also developed. Differentiable functions of a single complex variable are then considered. This study reveals a deep and fundamental theory whose development, by some of the giants of mathematics, such as Euler, Gauss, Riemann and Cauchy, began at the end of the 18th century. This surprisingly elegant branch of mathematics, known as Complex Analysis, has many dramatic applications across mathematics, engineering and the physical sciences.
Learning Outcomes
By the end of the module students should be able to:
Prove and apply a range of theorems concerning continuity and differentiability of real functions.
Understand the Riemann integral, and state and prove the fundamental theorem of calculus.
Understand and apply the basic theory of holomorphic functions of a complex variable.
Understand and apply fundamental theorems about complex variable functions such as Cauchy's Theorem, Cauchy's Integral Formula, and the Residue Theorem.
Evaluate Taylor series and Laurent series of complex-valued functions
Identify the poles and calculate the residues of complex valued functions
Apply techniques to evaluate contour in integrals and apply the theory of residues.
Assessment
Assessment Methods & Exceptions
Assessment:
2hr examination (80%) In-course assessment (20%) (including a variety of assessment possibly including problem sheets, class tests, online quizzes and group projects)
