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Complex Functions and Applications

Complex Functions and Applications

Code	14769
Year	3
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	Real Analysis and linear algebra
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.
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.
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.
Language	Portuguese. Tutorial support is available in English.