Instance: 2018/2019 - 2S
Cycles of Study/Courses
Teaching Staff - Responsibilities
Teaching - Hours
|Theoretical and practical :
Apprehension of the concepts and mastery of the techniques related to the analytical functions of a complex variable. Distinction between the real case and the complex case.
Studies of methods of integration, extension of definitions of the various elementary functions to the case of complex variable. Studies some applications in other areas of Mathematics and Natural Sciences. Required knowledge of Real Analysis and Linear Algebra.
Learning outcomes and competences
Assimilate the objectives defined in the previous paragraph.
Complex numbers: Algebraic and geometric representation of complex numbers. Riemann sphere and stereographic projection.
Elementary functions: Polynomial, rational function, root function, exponential function, logarithm, trigonometric functions, inverse trigonometric functions.
Complex sequences: Limits of successions of complex numbers, Bolzano-Weierstrass's theorem, Cauchy criterion.
Complex functions: Curves, plane sets, limit and continuity.
Holomorphic functions: Derivative and differential, Cauchy-Riemann equations.
Integral: Integral along a curve, primitive, Cauchy's theorem for continuously differentiable functions.
Cauchy integral formula and fundamental theorems of Complex Analysis: Cauchy integral formula, Cauchy inequality, Liouville's theorem, fundamental algebra theorem, Morera's theorem.
Analytic functions: Power series, uniform convergence, radius of convergence, Abel's theorem, Taylor series, uniqueness theorem.
Laurent's series and singularities: Laurent's theorem, classification of singular points.
Residuals: Residual calculus, residual theorem and applications, logarithmic residue, Rouché's theorem. Principle of isolated zeros, module theorem. Principle of analytic continuation. Analytic continuation of elementary functions. Integrate with parameters. Analytical properties of the integrals. Laplace, Fourier and Mellin transforms. Euler gama- function. Analytic continuation of the gama- function. Functions meromorfas and intéiras. Weierstrass Theorem: any continuous function is a uniform limit of polynomial functions.
1. G.Smirnov 'Complex Analysis and Applications', Escolar Editora, Lisbon, 2003.
2. Coimbra de Matos, José Carlos Santos, 'Course of Complex Analysis', Escolar Editora, Lisboa, 2000.
4. S. Lang, 'Complex Analysis', Springer-Verlag, 1999.
5. L.V. Ahlfors, 'Complex Analysis', McGraw-Hill, 1979.
G.Smirnov; Análise Complexa e Aplicações, Escolar Editora, Lisboa, 2003.
Teaching methods and learning activities
Exposition of the subject in theoretical classes, being completed by the resolution of exercises.
Physical sciences > Physics > Mathematical physics
Physical sciences > Mathematics > Mathematical analysis
Evaluation with final exam
Amount of time allocated to each course unit
|Frequência das aulas
Eligibility for exams
Calculation formula of final grade
final exam: 100%