Functional and Complex Analysis
- Unit Coordinator: Mariapia Palombaro, Gennaro Ciampa
- ECTS Credits: 9
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
Knowledge of basic topics of Functional Analysis, functional spaces and Lebesgue integral.
Knowledge of basic topics of complex analysis: elementary functions of complex variable, differentiation, integration and main theorems on analytic functions. Ability to use such knowledge in solving problems and exercises. - Content:
- Metric spaces, normed linear spaces.
- Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
- Crash course on Lebesgue measure and integration. Limit exchange theorems. Lp spaces. Hilbert spaces.
- Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators.
- Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Harmonic functions.
- Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liouville's theorem. Morera's theorem.
- Series representation of analytic functions. Taylor's theorem. Laurent's series and classiScation of singularities.
- Calculus of residues. The residue theorem. Application in evaluation of integrals on the real line and Principal Value. The logarithmic residue, Rouche's theorem.
- Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.
- Laplace transform and applications.
- Pre-requisites:
Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real functions of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations
- Reading list:
- J.E. Marsden, M.J. Hoffman, Basic complex analysis , Freeman New York. -
- W. Rudin, Real and complex analysis , Mc Graw Hill.
- Additional info:
Teaching methods
Lectures and tutorials
Assessment methods
Written and possibly oral exam
