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
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
- 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 classification 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.
- J.E. Marsden, M.J. Hoffman, Basic complex analysis , Freeman New York.
- W. Rudin, Real and complex analysis , Mc Graw Hill.