| Code |
14769
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| Year |
3
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| Semester |
S1
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| ECTS Credits |
6
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| Workload |
TP(60H)
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| Scientific area |
Mathematics
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Entry requirements |
Real Analysis and linear algebra
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Learning outcomes |
On completion of this unit, students will be able to:
1. understand the basic concepts of the theory of complex functions of one complex variable;
2. demonstrate the basic results of this theory;
3. apply the methods of complex analysis to evaluate real integrals;
4. understand the basic concepts and properties of Fourier series, Fourier transforms and Laplace transforms;
5. apply the methods of complex analysis to compute the integral (Fourier and Laplace) transforms and their inverses.
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Syllabus |
1. Holomorphic functions: the complex plane; complex-valued functions; continuity; differentiability and holomorphicity; elementary functions; conformal mapping and harmonic functions.
2. Complex integration: complex integrals; homotopy; fundamental theorem of calculus; Cauchy-Goursat theorem; Cauchy’s integral formula; Morera’s theorem; Liouville’s theorem.
3. Complex series: convergence of sequences, series and power series; Taylor series and Laurent series; uniqueness of series representation.
4. Residues calculus: zeros and poles; residues; the residue theorem; Rouché’s theorem; evaluation of improper real integrals.
5. Integral transforms: Fourier series; Fourier transforms; Laplace transforms.
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Main Bibliography |
1. The Fundamentals of complex analysis. E.B. Saff and A.D. Snider. 3rd edition, Prentice Hall, 2003.
2. Complex Analysis. John M. Howie. Springer Undergraduate Mathematics Series. 2008.
3. Complex Analysis. L. Ahlfors. 3rd edition, McGraw-Hill, 1979.
4. Análise de Fourier e Equações Diferenciais Parciais, D. G. de Figueiredo. 5ª edição, Impa, 2018.
5. Operational Mathematics, R. V. Churchill, McGraw-Hill, 3rd edition, 1971.
6. A First Course in Complex Analysis, version 1.54; M. Beck, G. Marchesi, D. Pixton, L. Sabalka.
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Teaching Methodologies and Assessment Criteria |
1. The teaching methodology is based on theoretical-practical lessons. The theoretical part is based on the teacher's presentation of the syllabus contents, based on the bibliography of the unit or other notes available. Great focus will be given to the rigorous demonstration of the main results. The practical part of the classes is based on solving exercises, both in an accompanying and autonomous way.
2. The assessment is done through two written tests, carried out in the middle and at the end of the semester, and exercises sheets for home resolution, which must be delivered by the students on dates previously fixed by the teacher. The final classification will be given by weighting the classifications of these evaluation elements, to be defined by the teacher at the beginning of the semester.
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Language |
Portuguese. Tutorial support is available in English.
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