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
scheda docente
materiale didattico
II. Cauchy's theorem and its applications: Goursat's theorem. Cauchy's theorem on starred sets. Cauchy's formula and residue calculus. Analytic continuation. Morera's theorem. Schwarz's principle.
III. Meromorphic functions and the logarithm: Zeros, poles, essential singularities. Meromorphic functions. Argument principle. Homotopy. The complex logarithm. Cauchy's theorem on simply connected regions.
IV. Canonical sums and products: Laurent series. Fourier-Laurent-Weierstrass theorem. Partial fractions; Mittag-Leffler theorem. Canonical Products and Weierstrass's Theorem.
V. Conformal Transformations: Elementary Conformal Maps and Fractional Linear Transformations (Möbius); Circle Automorphisms. Montel's Theorem and the Riemann Map Theorem.
Mutuazione: 20410882 AC310 - ANALISI COMPLESSA in Matematica L-35 R CHIERCHIA LUIGI
Programma
I. Preliminaries: Complex numbers and the complex plane. Topology and convergence. Continuous functions. Holomorphic functions and Cauchy-Riemann equations. Power series (Cauchy-Hadamard formula). Integration along curves.II. Cauchy's theorem and its applications: Goursat's theorem. Cauchy's theorem on starred sets. Cauchy's formula and residue calculus. Analytic continuation. Morera's theorem. Schwarz's principle.
III. Meromorphic functions and the logarithm: Zeros, poles, essential singularities. Meromorphic functions. Argument principle. Homotopy. The complex logarithm. Cauchy's theorem on simply connected regions.
IV. Canonical sums and products: Laurent series. Fourier-Laurent-Weierstrass theorem. Partial fractions; Mittag-Leffler theorem. Canonical Products and Weierstrass's Theorem.
V. Conformal Transformations: Elementary Conformal Maps and Fractional Linear Transformations (Möbius); Circle Automorphisms. Montel's Theorem and the Riemann Map Theorem.
Testi Adottati
Elias M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press, 2003Modalità Frequenza
Attendance is optional, and understanding of the textbook is sufficient for full enjoyment of the course. Attendance is naturally encouraged and STRONGLY recommended, as the interaction between instructor and students is a fundamental and unique learning tool.Modalità Valutazione
Assessment is based on a written exam and an oral exam. Two ongoing written exams are scheduled, which, if passed, replace the final written exam. Examples of past exams will be available online on the course website, which will be constantly updated by the instructor.
scheda docente
materiale didattico
II. Cauchy's theorem and its applications: Goursat's theorem. Cauchy's theorem on starred sets. Cauchy's formula and residue calculus. Analytic continuation. Morera's theorem. Schwarz's principle.
III. Meromorphic functions and the logarithm: Zeros, poles, essential singularities. Meromorphic functions. Argument principle. Homotopy. The complex logarithm. Cauchy's theorem on simply connected regions.
IV. Canonical sums and products: Laurent series. Fourier-Laurent-Weierstrass theorem. Partial fractions; Mittag-Leffler theorem. Canonical Products and Weierstrass's Theorem.
V. Conformal Transformations: Elementary Conformal Maps and Fractional Linear Transformations (Möbius); Circle Automorphisms. Montel's Theorem and the Riemann Map Theorem.
Mutuazione: 20410882 AC310 - ANALISI COMPLESSA in Matematica L-35 R CHIERCHIA LUIGI
Programma
I. Preliminaries: Complex numbers and the complex plane. Topology and convergence. Continuous functions. Holomorphic functions and Cauchy-Riemann equations. Power series (Cauchy-Hadamard formula). Integration along curves.II. Cauchy's theorem and its applications: Goursat's theorem. Cauchy's theorem on starred sets. Cauchy's formula and residue calculus. Analytic continuation. Morera's theorem. Schwarz's principle.
III. Meromorphic functions and the logarithm: Zeros, poles, essential singularities. Meromorphic functions. Argument principle. Homotopy. The complex logarithm. Cauchy's theorem on simply connected regions.
IV. Canonical sums and products: Laurent series. Fourier-Laurent-Weierstrass theorem. Partial fractions; Mittag-Leffler theorem. Canonical Products and Weierstrass's Theorem.
V. Conformal Transformations: Elementary Conformal Maps and Fractional Linear Transformations (Möbius); Circle Automorphisms. Montel's Theorem and the Riemann Map Theorem.
Testi Adottati
Elias M. Stein, R. Shakarchi, Complex Analysis, Princeton University Press, 2003Modalità Frequenza
Attendance is optional, and understanding of the textbook is sufficient for full enjoyment of the course. Attendance is naturally encouraged and STRONGLY recommended, as the interaction between instructor and students is a fundamental and unique learning tool.Modalità Valutazione
Assessment is based on a written exam and an oral exam. Two ongoing written exams are scheduled, which, if passed, replace the final written exam. Examples of past exams will be available online on the course website, which will be constantly updated by the instructor.