Complex Analysis

Additional Info

  • ECTS credits: 6
  • Semester: 2
  • University: University of L'Aquila
  • Prerequisites:

     

    Knowledge of all topics treated the Mathematical Analysis courses in the first and second year: real function of real variables, limits, differentiation, integration; sequences and series of funcions; ordinary differential equations.

  • Objectives:

     

    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

  • Topics:

    Complex numbers. Sequences. Elementary functions of complex numbers. Limits, continuity. Differentiation. Analytic functions. Armonic functions.

    Contour integrals. Cauchy's Theorem. Cauchy's integral formula. Maximum modulus theorem. Liuville'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. Rouche's theorem.

    Conformal mappings. Main theorems. Fractional linear transformations.

    Fourier transform for L^1 functions. Applications. Fourier transform for L^2 functions. Plancherel theorem.

    Laplace transform and applications.

  • Books:

    J.E. Marsden, M.J. Hoffman, Basic complex analysis. Freeman New York.

    W. Rudin, Real and complex analysis. Mc Graw Hill.

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