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Module Title LI Real & Complex Analysis Mathematics Mathematics 06 25666 Dr Chris Good Intermediate Level 20 Full Term None 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. By the end of the module students should be able to: understand the concepts and properties of limits, continuity and differentiability for real functionsevaluate limits and derivatives for examples involving well-known functionsprove and apply theorems concerning continuity and differentiability, such as the Intermediate Value Theorem, the Mean Value Theorem and Taylor’s Theoremdefine the integral of a real valued function on a closed bounded integral and apply this definition in simple situationsstate and prove the fundamental theorem of calculusevaluate Taylor series and Laurent series of complex-valued functionsidentify the poles and calculate the residues of complex valued functionsstate Cauchy’s Integral theorem and the residue theorem and use them to evaluate real integralsdefine and give examples of certain topological properties of Euclidean space, such as open, closed and compact sets 25666-01 : Raw Module Mark : Coursework (100%) Assessment: 3 hour examination (80%), work done during semester (20%)Reassessment: best of 3 hour resit examination (100%) or 3 hour resit examination (80%) and work done during the semester (20%)