Complex Analysis
Code
7813
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Luís Manuel Trabucho de Campos
Weekly hours
5
Total hours
70
Teaching language
Português
Prerequisites
Working knowledge of real analysis (one and several variables), analytic geometry of the plane and the usual topology of R2.
Subject matter
1. Complex Functions. Algebra of complex numbers.Definition of the elementary complex functions. Limits and continuity. Differentiability - analytic functions. Harmonic functions. Differentiability of the elementary functions. Conformal mappings; fractional linear transformations
2. Complex integration - Cauchy’s Theorem and applications.Complex integration. Cauchy’s Theorem. Cauchy’s Integral Formula. Fundamental theorems: Morera’s theorem, Cauchy’s inequalities, Liouville’s theorem, Fundamental Theorem of Algebra, maximum principle.
3. Power series; Laurent series. Pointwise and uniform convergence of function sequences and series. Power series.Taylor’s Theorem; analyticity. Singularities – Laurent series. Isolated singularities; classification of isolated singularities
4. Residues. Calculation of residues. Residue theorem. Evaluation of definite integrals.
5. Conformal Mapping. Exemples and applications.
Bibliography
L. V. Ahlfors, Complex Analysis, McGraw-Hill (1979)
M. A. Carreira e M. S. Nápoles, Variável complexa - teoria elementar e exercícios resolvidos, McGraw-Hill (1998)
S. Lang, Complex Analysis, Springer (1999), ISBN 0-387-98592-1
J. E. Marsden and M. J. Hoffman, Basic Complex Analysis - Third Edition, Freeman (1999), ISBN 0-7167-2877-X
Teaching method
The theory is explained and illustrated with examples. Main results are proved. The students are given the opportunity of working in a list of problems, with the instructor´s support if needed, and the instructor´s comments on relevant results highlighted in the problems.
Evaluation method
1. The assessment will be done through three tests (T1, T2, T3).
2. The tests will all have the same weight. Each test will be graded 0-20 values.
3. To pass, students must obtain a final classification equal to, or higher than 10. The final classification (CF) is obtained by rounding the following value (V), to the units
V = (C1 + C2 + C3) / 3,
where C1, C2 and C3 represent the grades of tests T1, T2, T3, respectively.
Example: if V = 12.4 then CF = 12; if V = 12.5 is then CF = 13.
4. An exam will take place after the end of the course for those who did not succeed. The classification will be an integer from 0 to 20. The student will pass if the classification is 10 or higher, in a maximum of 20.