D1. To know the fundamental results on measure theory, integration and Lebesgue spaces for Mod. A; the fundamental results on complex analysis and the first elements of harmonic analysis for Mod. B
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 presentation.
D5. To be able to autonomously consult the specialized texts.

Differential calculus and integral calculus on R^N, metric spaces.
Complex numbers, Power series, intecration along curves and differential
forms in the plane.

Mod. A Measure theory. Integration. Spaces of integrable functions. Mod. B Holomorphic functions of a complex variable. Harmonic functions in the plane.

Mod. A
E.H.Lieb, M. Loss, Analysis, American Mathematical Society, 1997
H.L. Royden, Real Analysis, MacMillan, 1968

Mod. B
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

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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

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Mod. A Oral examinations. The oral examination aims to access the students’ knowledge of the theoretical and the applicative aspects of the covered topics. Mod. B 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. The final mark will be the arithmetic mean of the partial marks,, approximated by excess.