Complex Analysis

A.Y. 2019/2020
9
Max ECTS
64
Overall hours
SSD
MAT/05
Language
Italian
Learning objectives
The course aims at providing some basic concepts and results in study of holomorphic functions in one complex variable.
Expected learning outcomes
At the end of the course, students will acquire the basic knowledge of holomorphic function theory in one complex variable, and they will be able to apply it to exercises that need also computational techniques
Course syllabus and organization

Single session

Responsible
Lesson period
Second semester
Prerequisites for admission
Analisi Matematica 1, 2, 3 and 4.
Assessment methods and Criteria
The final examination consists of a written exam and of an oral exam, given on the same day.

- During the written part, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in complex analysis. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 3 or 4 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral part of the exam, the student will be required to illustrate results presented during the course and will be required to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if both the written and oral parts are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Complex Analysis (first part)
Course syllabus
unzioni olomorfe, equazioni di Cauchy-Riemann.
Integrali di linea, primitive olomorfe. Il teorema di Cauchy e la formula integrale di Cauchy. Serie di potenze e loro proprietà.
Regolarità delle funzioni olomorfe: il teorema degli zeri, il principio di identità.
Conseguenza della formula integrale di Cauchy: il teorema di Weierstrass, formula delle derivate e principio del massimo modulo.
Teorema della mappa aperta, dell'invertibilità locale di una funzione olomorfa Teorema globale di Cauchy.
Singolarità isolate e sviluppi di Laurent. Calcolo dei residui e applicazioni.
Funzioni armoniche. Integrale di Poisson.
Il teorema di Rouchè e il teorema dell'indicatore logaritmico.
Il lemma di Schwarz e il gruppo degli automorfismi del disco.

[Materiale di riferimento]:
Appunti del corso in rete
- S. Lang, Complex Analysis, 4th Edition, Springer-Verlag Ed.
- R. Churchill and J. Brown, Complex Variables and Applications, McGraw-Hill Inc.
- J.B. Conway, Functions of One Complex Variable I, , Springer-Verlag Ed.
- E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Univ. Press
- L. Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill Science Ed

[Program with reference to descriptor 1 and 2]:
-Holomorphic functions, Cauchy-Riemann equations.
-Line integrals, holomorphic anti-derivatives. Cauchy theorem and Cauchy integral formula. Power series and their properties.
-Regularity of holomorphic functions: zeros of holomorphic functions,
the identity principle and the maximum modulus theorem, Schwarz lemma.
-Consequences of Cauchy integral formula: Weierstrass theorem, Cauchy formula for the derivatives and the maximum modulus principle.
-Isolated singularities and Laurent expansion.
-Calculus of residues and applications to definite integrals.
Rouche' theorem and the argument principle.
-Open mapping theorem and local invertibility of a holomorphic function.
-Harmonic functions, Poisson integral.
Teaching methods
Traditional blackboard lectures. Attandance strongly suggested.
Teaching Resources
- M. Peloso, Course's notes, available online.
- S. Lang, Complex Analysis, 4th Edition, Springer-Verlag Ed.
- R. Churchill and J. Brown, Complex Variables and Applications, McGraw-Hill Inc.
- J.B. Conway, Functions of One Complex Variable I, , Springer-Verlag Ed.
- E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Univ. Press
- L. Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill Science Ed
Complex Analysis (mod.02)
Course syllabus
Linear fractional transformations. Riemann mapping theorem.
Infinite products. Entire functions. Weierstrass and Hadamard factorization theorems.
Analytic continuation.
Euler's gamma function and Riemann's zeta function.
Teaching methods
Traditional blackboard lectures. Attandance strongly suggested.
Teaching Resources
- M. Peloso, Appunti del corso in rete
- S. Lang, Complex Analysis, 4th Edition, Springer-Verlag Ed.
- R. Churchill and J. Brown, Complex Variables and Applications, McGraw-Hill Inc.
- J.B. Conway, Functions of One Complex Variable I, , Springer-Verlag Ed.
- E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Univ. Press
- L. Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill Science Ed
Complex Analysis (first part)
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 10 hours
Guided problem-solving: 6 hours
Lessons: 28 hours
Complex Analysis (mod.02)
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 3
Guided problem-solving: 6 hours
Lessons: 14 hours
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica