Course Details for A.Y. 2019/2020

Course 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

Course Content

• 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. Liouville's theorem. Morera 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.

Prerequisites and Learning Activities

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

Assessment Methods and Criteria

Written exam and oral exam

Textbooks

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

Course page updates

• A.Y. 2015/2016
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• A.Y. 2019/2020