To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals.
Curriculum
teacher profile teaching materials
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Mutuazione: 20410882 AC310 - ANALISI COMPLESSA in Matematica LM-40 CHIERCHIA LUIGI
Programme
I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
Core Documentation
[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Type of delivery of the course
Lectures and exercises with the desirable active participation of students.Attendance
Attendance in class is strongly recommended: for the optimal performance of the course, active interaction between students and teacher is of fundamental importance.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. teacher profile teaching materials
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Mutuazione: 20410882 AC310 - ANALISI COMPLESSA in Matematica LM-40 CHIERCHIA LUIGI
Programme
I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
Core Documentation
[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Type of delivery of the course
Lectures and exercises with the desirable active participation of students.Attendance
Attendance in class is strongly recommended: for the optimal performance of the course, active interaction between students and teacher is of fundamental importance.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher. teacher profile teaching materials
II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Mutuazione: 20410882 AC310 - ANALISI COMPLESSA in Matematica LM-40 CHIERCHIA LUIGI
Programme
I. Elementary theory (Including: Complex numbers and the complex plane. Convergence. Sets in the complex plane. Functions on the complex plane. Continuous functions. Holomorphic functions. Power series. Integration along curves).II. Cauchy's theorem and its applications (Including: Goursat's theorem; Cauchy's formula and calculation of residues. Analytical continuation. Morera's theorem. Schwarz's principle). Cauchy's theorem in simply connected domains.
III. Meromorphic functions and the logarithm (Including: zeros and poles; isolated singularities. Argument principle. Rouché's theorem).
IV. Conformal transformations (Including: elementary maps and linear fractional transformations); Riemann mapping theorem.
V. Laurent series; partial fractions and canonical products.
Core Documentation
[S] Complex Analysis. Elias M. Stein, Rami Shakarchi Princeton University Press 2003, ISBN 10: 1400831156 / ISBN 13: 9781400831159[A] Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1
[E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.
Type of delivery of the course
Lectures and exercises with the desirable active participation of students.Attendance
Attendance in class is strongly recommended: for the optimal performance of the course, active interaction between students and teacher is of fundamental importance.Type of evaluation
The evaluation is based on a written test and an oral exam. There are two written tests "in progress" which, in the event of a positive outcome, replace the final written test. Examples of tests of past years will be available on the web site dedicated to the course which will be constantly updated by the teacher.