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

Review of Fourier series. Well-posed problems for the heat equation. Cauchy-Dirichlet problem for the one-dimensional heat equation; method of separation of variables. Uniqueness. Well-posed problems for the wave equation. Cauchy-Dirichlet problem for the one-dimensional wave equation; method of separation of variables. D’Alembert’s formula; uniqueness and continuous dependence on the data for solutions of Cauchy problems for the one-dimensional wave equation. Well-posed problems for the Poisson equation. Dirichlet problem for the Laplace equation in the disc. Maximum principle. Uniqueness and continuous dependence on the data of the solution to the Dirichlet problem for the Poisson equation. Harmonic functions. Green’s identities; fundamental solution of the Laplace operator. Representation formulas. Green’s function in the ball. Poisson formula. Classification of second-order linear partial differential equations in two variables.

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.