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Faculdade de Ciências e Tecnologia

Complex Analysis



Academic unit

Faculdade de Ciências e Tecnologia


Departamento de Matemática



Teacher in charge

Bento José Carrilho Miguens Louro

Weekly hours


Total hours


Teaching language



Basic features of the theory of complex variable functions, the main goal being to obtain the equivalence of the four different characterizations of the concept of analytic function; other goals being some study of global type geometric properties of analytic functions, and some notions about isolated singularities. 


Working knowledge of real analysis (one and several variables), analytical 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.


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

In the lectures the theory is explained and illustrated with examples. Main results are proved.

In the problem solving sessions, 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

There are two tests that can substitute the final exam in case of approval. Otherwise the student must pass the final exam. More detailed rules are  available in the portuguese version.