This module starts by developing the theory of continuous and differentiable functions of one real variable introduced in Real Analysis and the Calculus. It places the familiar techniques of differentiation, such as the Chain Rule, on a firm theoretical foundation and proves some of the key results of real analysis such as the Intermediate Value Theorem, the Mean Value Theorem and Taylor’s Theorem. 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. It quickly provides us with powerful new techniques of integration and has far-reaching consequences in theoretical physics, electronics, fluid mechanics and thermodynamics.

Underlying topological properties of Euclidean space common to both real and complex analysis are mentioned throughout the module.

Learning Outcomes

By the end of the module students should be able to:

Understand the concepts and properties of limits, continuity and differentiability for real functions

Evaluate limits and derivatives for examples involving well-known functions

Prove and apply theorems concerning continuity and differentiability, such as the Intermediate Value Theorem, the Mean Value Theorem and Taylor’s Theorem

Define the integral of a real valued function on a closed bounded integral and apply this definition in simple situations

State and prove the fundamental theorem of calculus

Evaluate Taylor series and Laurent series of complex-valued functions

Identify the poles and calculate the residues of complex valued functions

State Cauchy’s Integral theorem and the residue theorem and use them to evaluate real integrals

Define and give examples of certain topological properties of Euclidean space, such as open, closed and compact sets

Students taking the module at Level H will explore the subject beyond the taught syllabus.