Subject: Complex Analysis

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - To be able to work with complex numbers.
2 - To know the fundamental results in the theory of complex valued functions of complex variable.
3 - To identify the similarities and the differences between the differential and integral calculus in the field of complex numbers and in R^2.
4 - To apply the theory of residues when solving certain problems.

# Syllabus:

1 - Complex Numbers.
1.1 - General Notions.
1.2 - De Moivre's Formula. Roots of Complex Numbers.
1.3 - Subsets of the Complex Plane.
2 - Complex Valued Functions of a Complex Variable.
2.1 - Exponential Function.
2.2 - Logarithmic Function.
2.3 - Trigonometric and Hyperbolic Functions.
2.4 - Complex Powers.
3 - Differential Calculus.
3.1 - Topological Notions.
3.2 - Limits and Continuity.
3.3 - Differentiability and the Cauchy-Riemann Equations.
3.4 - Derivatives of the Elementary Functions.
3.5 - Harmonic Functions.
4 - Integral Calculus.
4.1 - Line Integrals.
4.2 - Cauchy's Theorem.
4.3 - Cauchy's Integral Formulas.
4.4 - Applications of the Cauchy's Integral Formulas.
5 - Series Representation of Analytic Functions.
5.1 - Sequences and Series of Complex Numbers. Power Series.
5.2 - Analytic Functions and Taylor's Theorem.
5.3 - Laurent Series and Classification of Singularities.
6 - Residues.
6.1 - Calculation of Residues.
6.2 - Residue Theorem.
6.3 - Applications of the Residue Theorem.

# Literature/Sources:

Barreira, L. , 2009 , Análise Complexa e Equações Diferenciais , IST Press
Hoffman, M. J. and Marsden, J. E. , 1987 , Basic Complex Analysis , W. H. Freeman and Company
Rudin, W. , 1987 , Real and Complex Analysis, 3rd Ed. , McGraw-Hill Book Company
Abreu, António Simões , 2009 , Funções de Variável Complexa, Teoria e Aplicações , IST Press

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
The lectures are used to introduce the concepts, results and methods to the student. In the problem solving classes, the student is expected to solve problems from the problem sets given, individually or in a group. In the normal season, the evaluation is composed by two compulsory written tests, each corresponding to 50% of the final grade. In the appeal season, students who have failed to pass the course during the normal season can choose between retaking one or the two tests, each corresponding to 50% of the final grade.