D1. To know the fundamental results on complex analysis and the first elements of harmonic analysis.
D2. To apply the theoretical acquired skills to solve problems and exercises.
D3. To recognize the basic techniques of the covered topics for their applications to new problems.
D4. To be endowed with the competence to express the fundamental concepts with command of the language and a proper presenation.
D5. To be able to autonomously consult the specialized texts.

Complex numbers, Power series, intecration along curves and differential forms in the plane.

Holomorphic functions of a complex variable. Cauchy-Riemann equations. Cauchy identity and Cauchy formulae. Maximum modulus principle. Liouville Theorem. Fundamental Theorem of Algebra. Taylor series and Laurent series. Isolated singularities of Holomorphic functions. Index of a curve with respect to a point. Residue theorem, argument principle and Rouche Theorem. Harmonic functions and conformal maps.

W. Rudin, Real and complex analysis, Second edition. McGraw-Hill, 1974.

L.V. Ahlfors, Complex Analysis, McGraw-Hill, 1953.

B.P. Palka, An introduction to complex function theory, Springer, 1991.

Holomorphic functions of a complex variable. Cauchy-Riemann equations. Cauchy identity and Cauchy formulae. Maximum modulus principle. Liouville Theorem. Fundamental Theorem of Algebra. Taylor series and Laurent series. Isolated singularities of Holomorphic functions. Index of a curve with respect to a point. Residue theorem, argument principle and Rouche Theorem. Calculus with residues.Chains and cycles. Cauchy theorem. Homology of an open set in the plane. Riemann mapping theorem. Harmonic functions. Dirichlet problem.

Lectures at the blackboard, consisting in exposing theoretical contents and solving a certain amount of practical exercises. Other exercises will be assigned for personal or group homework, as fundamental part of individual study.

.

Written and oral exam. The written part consists in solving exercises and problems concerning the topics treated in the lectures. The student who gets a mark lower than 15/30 in the written proof is adviced not to try the oral exam. The oral exam covers the theoretical aspects treated in the lectures, and possibly some more exercises. In order to achieve a final mark of 18/30 the student must demonstrate a sufficient basic knowledge of all the topics treated in the corse. In order to achieve the highest mark of 30/30 (possibly cum laude) the student must demonstrate an accurate and deep knowledge of all the topics treated in the course.