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Complex Analysis

Code: M2008     Acronym: M2008     Level: 200

Classification Keyword
OFICIAL Mathematics

Instance: 2018/2019 - 2S

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: First Degree in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
L:B 0 Official Study Plan 3 - 6 56 162
L:CC 0 Plano de estudos a partir de 2014 2 - 6 56 162
L:F 26 Official Study Plan 2 - 6 56 162
L:G 0 study plan from 2017/18 2 - 6 56 162
L:M 114 Official Study Plan 2 - 6 56 162
L:Q 5 study plan from 2016/17 3 - 6 56 162

Teaching Staff - Responsibilities

Teacher Responsibility
Semyon Borisovich Yakubovich

Teaching - Hours

Theoretical classes: 2,00
Theoretical and practical : 2,00
Type Teacher Classes Hour
Theoretical classes Totals 1 2,00
Semyon Borisovich Yakubovich 2,00
Theoretical and practical Totals 3 6,00
Semyon Borisovich Yakubovich 6,00

Teaching language



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.

Working method



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.

Mandatory literature

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 Type

Evaluation with final exam

Assessment Components

designation Weight (%)
Exame 100,00
Total: 100,00

Amount of time allocated to each course unit

designation Time (hours)
Estudo autónomo 100,00
Frequência das aulas 56,00
Trabalho escrito 6,00
Total: 162,00

Eligibility for exams


Calculation formula of final grade

final exam: 100%
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