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Department of Mathematics and Applications
"Renato Caccioppoli"


Complements of Mathematical Analysis

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Complements of Mathematical Analysis

Credits

6

Prerequisites

Mathematical Analysis 2.

Scientific-disciplinary sector (SSD)

MAT/05 Mathematical Analysis.

Examination method

Oral exam.

Learning
objectives

The course aims to provide an introduction to the theory of functions of a complex variable. Analytic and geometric properties of functions of a complex variable are developed, with particular regard to applications in the theory of partial differential equations, such as the Laplace equation.

Syllabus

Algebraic, geometric and exponential form of a complex number. nth roots of a complex number. Holomorphic functions and Cauchy-Riemann equations. Analyticity of holomorphic functions. Laurent series expansions and classification of isolated singular points. The residue theorem. Zeros of a holomorphic function. Open mapping theorem. Mean value property and maximum modulus principle. Computation of integrals of real-valued functions by means of the residue theorem. Conformal maps. Theory of distributions. Fourier series expansion. Properties of harmonic functions – Maximum principle and mean value property. Fundamental solution of the Laplace operator. Green’s identities. The Green function. The Dirichlet problem for the Laplace equation in the ball: Poisson integral formula. Characterisation of harmonic functions. Solution of the Dirichlet problem for the Laplace equation in simply connected plane domains.

Expected learning
outcomes

At the end of the course, students must demonstrate that they

  • know and understand the general issues related to functions of a complex variable;
  • are able to apply the knowledge acquired to the study of some partial differential problems, with particular regard to the Laplace equation;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.