Complex Analysis
A.Y. 2024/2025
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
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
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.
- 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 is 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 2 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 is required to illustrate results presented during the course and to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capability 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 are communicated after the oral examination.
- 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 is 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 2 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 is required to illustrate results presented during the course and to solve problems in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capability 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 are communicated after the oral examination.
Complex Analysis (first part)
Course syllabus
-Holomorphic functions, Cauchy-Riemann's equations.
-Line integrals, holomorphic anti-derivatives. Cauchy's theorem and Cauchy's integral formula. Power series and their properties.
-Regularity of holomorphic functions: zeroes of holomorphic functions,
the identity principle and the maximum modulus theorem, Schwarz's lemma.
-Consequences of Cauchy's integral formula: Weierstrass theorem, Cauchy's formula for the derivatives and the maximum modulus principle.
-Isolated singularities and Laurent's expansion.
-Calculus of residues and applications to definite integrals.
Rouché's theorem and the argument principle.
-Open mapping theorem and local invertibility of a holomorphic function.
-Harmonic functions, Poisson integral.
-Line integrals, holomorphic anti-derivatives. Cauchy's theorem and Cauchy's integral formula. Power series and their properties.
-Regularity of holomorphic functions: zeroes of holomorphic functions,
the identity principle and the maximum modulus theorem, Schwarz's lemma.
-Consequences of Cauchy's integral formula: Weierstrass theorem, Cauchy's formula for the derivatives and the maximum modulus principle.
-Isolated singularities and Laurent's expansion.
-Calculus of residues and applications to definite integrals.
Rouché's theorem and the argument principle.
-Open mapping theorem and local invertibility of a holomorphic function.
-Harmonic functions, Poisson integral.
Teaching methods
Traditional blackboard lectures. Attendance is strongly suggested.
Teaching Resources
- M. Peloso, Personal notes.
- 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
- 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. Euler's Gamma function.
Infinite products. Entire functions. Weierstrass and Hadamard factorization theorems. Euler's Gamma function.
Teaching methods
Traditional blackboard lectures. Attendance strongly suggested.
Teaching Resources
- M. Peloso, Notes
- 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
- 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: 24 hours
Lessons: 28 hours
Lessons: 28 hours
Professors:
Terraneo Elide, Vignati Marco
Complex Analysis (mod.02)
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 3
Lessons: 21 hours
Professor:
Vignati Marco
Professor(s)
Reception:
Wed, 12.30 am - 2.00 pm; otherwise, contact me via e-mail
Math Dept., via C.Saldini 50, room R013, ground floor