Complex Analysis
A.Y. 2018/2019
Learning objectives
Introduction to some basic concepts and results in study of holomorphic functions in one complex variable. The following topics are considered. Complex integration and Cauchy's theorem, Cauchy's integral formula and its consequences. Singularities, residues and applications. Conformal mappings and the Riemann mapping theorem. Entire functions and their factorizations. Harmonic fiunctions in the disc and Poisson's formula.
Expected learning outcomes
Knowledge of the topics and results, and application to exercises that need also computational techniques.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Complex Analysis (first part)
Course syllabus
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.
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.
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.
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.
Complex Analysis (mod.02)
Course syllabus
-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.
-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.
Complex Analysis (first part)
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 10 hours
Guided problem-solving: 6 hours
Lessons: 28 hours
Guided problem-solving: 6 hours
Lessons: 28 hours
Professors:
Monguzzi Alessandro, Peloso Marco Maria
Complex Analysis (mod.02)
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 3
Guided problem-solving: 6 hours
Lessons: 14 hours
Lessons: 14 hours
Professors:
Monguzzi Alessandro, Peloso Marco Maria
Professor(s)
Reception:
By appointment
My office, room 1021 Dipartimento di Matematica