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Content

This course provides an introduction to the theory and application of complex variables and complex functions. The properties of elementary complex functions are outlined, and the concept of analyticity is developed in its entirety. The most fundamental theorems are stated, proved and utilized throughout. Particular emphasis is placed on the development of integral calculus in the complex plane. Practice problems will be incorporated throughout to provide concrete examples of how to apply the theory. Review of complex numbers: - Algebraic and geometric representation of complex numbers; - Euler’s formula; - Rational powers and roots of complex numbers; - Regions in the complex plane. Analytic functions: - Limits, continuity and differentiability; - Cauchy Riemann equations; - Analytic and harmonic functions; - Introduction to conformal mapping. Elementary functions: - The complex exponential function; - Trigonometric and Hyperbolic functions and inverses; - The complex logarithm – definition, properties, branches and branch cuts. Complex Integration: - Statement and proof of Cauchy’s Fundamental Theorem, Cauchy’s Integral Formulae (for simply-connected and multiply-connected regions); - Evaluation of contour integrals via the Cauchy theorems and Cauchy’s integral formulae; - Statement and proof of Taylor’s Theorem and Laurent’s Theorem. Series: - Convergence of sequences and series; - Power series – absolute and uniform convergence, integration and differentiation; - Taylor and Laurent series. Residues and Poles: - Isolated singular points, residues and the Residue Theorem; - Classifying isolated singular points; - Residues at poles; - Evaluation of improper real integrals by contour integration around poles.

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Conditions of Participation

A sound knowledge of introductory real analysis is required. For this reason, MATH 2277: Introduction to Real Analysis I (or equivalent) is listed as a course prerequisite.

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Teaching Methods

- The online teaching tool, myeLearning, will be used during this course for communication among students and staff for official posting of important notices, provision of recommended resource materials and links to resources on specific websites. - This course will be delivered through a combination of informative lectures/ tutorials. Supporting course materials will be posted on myeLearning. - Six practice problem sheets will be posted on myelearning, and solutions will be provided during class (*in the form of tutorial sessions). This is not to be graded.

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Learning outcomes / Competences

By the end of the course, students will be able to: - Prove that a complex function is continuous, differentiable or analytic at a point or given region of the complex plane. - Identify and construct analytic functions using the Cauchy Riemann equations. - Describe the properties of conformal mappings, and find basic Mobius transforms across complex planes. - Manipulate elementary complex functions (exponentials, trigonometric, logarithmic and hyperbolic functions). - State and prove Cauchy’s Fundamental Theorem, Cauchy’s Integral Formulae (for simply-connected and multiply-connected regions). - Evaluate contour integrals via the Cauchy theorems and Cauchy’s integral formulae. - State and prove Taylor’s Theorem and Laurent’s Theorem. - Investigate the convergence of a complex sequence. - Test a complex power series for absolute or uniform convergence. - Provide the Taylor or Laurent series representation for a complex function in a given region. - Classify isolated singular points and compute residues at poles. - Utilize the Residue Theorem to evaluate improper real integrals.

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Forms of examination

- Final Examination (50%) - Coursework Exams (40%) – two 20% coursework examinations will be given. - Coursework Assignments (10%) – two 5% assignments will be given. Examinations (coursework and final) and coursework assignments will be delivered online via the assignment tool from the moodle (myelearning) platform. Examinations will be in asynchronous/ open book format, and students will be given 24 hours to submit their scanned handwritten solutions. Assignments must be submitted within one week (7 days).

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Conditions for allocation of credit points

As is the case for all undergraduate courses at the UWI, passing grades will be given for an overall score (coursework and final exam components combined) of 50%.

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Other information

This is a core course for the BSc Mathematics (Special) degree programme at the UWI, St. Augustine, and it is an elective course for any other undergraduate degree programme.