teaching activity

Courses given in Italian

Math Method course - aa 2012-2013

On Friday, October 26 2012, Prof. Citti will teach

Exam:

• An (optional) middterm 1,30 hours : 4 exercices. topic covered by the midterm:
  • establish where a given function is holomorphic (see exercice 12 in notes)
  • Integrate the given functions (see exercice 4 pag 54 in notes)
  • use residues to compute integrals (see exercice 14 pag 114 in notes)
  • compute integrals on $\R$ using residues
• Students who passed the middlte term can come on the day of the oral to
  • solve an exercice on eigenvectors
  • answer an oral question typical questions
• Students who did not pass the middlte term will have to repeat a test on all the exercices
  • they can take a make up midterm, (1h + 30 minutes) and then afford the exercice on eigenfunctions on the day of the oral
  • alternatively they can choose afford all the exercices (the 4 exercices of the middle term and the one on eigenfunctions (= 2hours). In this case they will only discuss oral on the day of the oral

Dates:

• Dates of the written exam or mid-term recovery
  • 22 Dec. 2012, 9.00 AM, classroom 5.7
  • 25 Jan. 2013, 14. PM, classroom 6.2
  • 15 Feb. 2013, 14 PM, classroom 6.2.
• Dates for the oral exam:
  • 7 Jan. 2013, 9:00 AM, classroom "Tonelli" (Maths. Dept., 6th floor)
  • 31 Jan. 2013, 9:00 AM, classroom "Tonelli" (Maths. Dept., 6th floor)
  • 20 Feb. 2013, 9:00 AM, classroom 4.2

Program of the course

• Functions of complex variables and residue
• Introduction to couontable basis of an Hilbert space: Fourier analysis, Bessel Functions, Zernike Polynomials.

Notes and suggested reading

• Functions of complex variables and residue. You can as well use
  • Complex analysis
    or A first course in complex Analysis or the notes in the webpage of Prof. Arcozzi
• Detailed description of the topic afforded in class.
  Functions of complex variables
  • Graphs of functions of complex variables - some graphs
  • Notion of differentiability and Holomorphicity, The Cauchy Riemann Equations (see chapter 2 or complex derivatives)
  • Integration: definition and the Cauchy integral formula (see chapter 4 or line integrals)
  • Power series, region of convergence (see chapter 7 or series)
  • Zeros, pole and resudue (see chapter 9.1, 9.2 )
  • Fourier and Laplace transforms (see pag 18, )
  Hilbert spaces:
  • cap 1: vector spaces.
  • cap 2 Hilbert spaces.
  • cap 3 Fourier Analysis
  • Spectral theorem for self-adjoint compact operators
  • Zernicke polynomials

Suggested exercices

• Exercice 12, chapter 2, pag 26 (see solutions on pag 121) in the notes
• Exercices 4, 5 chapter 4 pag 54 in the notes
• Exercices in chapter 9 in the notes
• integrals
• eigenvectors
• questions for the oral exam

Results of the written tests

previous years program and notes

• a.a. 2011-2012
• a.a. 2010-2011
• a.a. 2009-2010