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Module Title Real and Complex Variable Theory Mathematics Mathematics 06 22497 A Sobolev Intermediate Level 20 Full Term none Lecture-46 hours Tutorial-10 hours Total: 56 hours This module follows on from the second half of material in prerequisites Foundation and Abstraction 1 and 2 and gives a rigorous treatment of limits, continuity and differentiability for real functions. The material formalises and justifies many of the basic theorems and techniques of earlier modules in calculus. The module then extends to complex-valued functions, defined on regions of the complex plane, all the ideas and techniques of the integral and differential calculus which have been studied previously. By the end of the module the student will 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; appreciate and apply theoriems concerned with continuity and differentiability (eg the Intermediate Value Theorem, the Mean Value Theorem and Taylor's Theorem); determine where a complex-valued function is analytic and identify its singular points; evaluate Taylor series and Laurent series of complex-valued functions; apply Cauchy's Integral theorem and the residue theorem to evaluate real integrals. 22497-01 : CA Sem 1 : Coursework (10%) 22497-02 : CA Sem 2 : Coursework (10%) 22497-03 : Exam : Exam (Centrally Timetabled) - Written Unseen (80%) 3 hr examination 80%, coursework and/or class test 20% none