Faculties and centres
This subject requires having successfully completed the subjects corresponding to Real Analysis, of one and several variables, and a course about Topology on sets. The main objective of this course is to provide basic training on the theory of functions of a complex variable (or also called Complex Analysis) through the Cauchy theory based on the notion of integral along a path, the Cauchy's global theorem and its applications to the theory of residue. Different applications and consequences of previous results, such as Rouché's theorem or open and inverse mapping theorems, as well as other specific results on power series and infinite products, will be topics to be conceptualized in this course.
Specific Competences (CE)
The main objective of this subject is to show the main topics of the Theory of Complex Variable Functions (also called Complex Variable or Complex Analysis). Therefore, basic operations with complex numbers, inequalities, geometric representations, and calculation of roots and logarithms will have to be mastered initially. From the concept of derivability, in this first part of the course, the student must assimilate the important notion of analytic function and its relation with Cauchy-Riemann equations. In this context, the treatment of elementary functions such as exponential, logarithmic, power, trigonometric and hyperbolic functions will also be required. The second part of the subject deals with the work process carried out in the Cauchy theory based on the integral along a path. The students must embrace this process. Different applications and consequences of the Cauchy theory, such as Morera's theorem, Schwarz's reflection principle, Liouville's theorem or maximum modulus principle, will be a primordial purpose in this second part of the subject. In addition, the third part deals with learning and knowing how to handle the applications of the Cauchy's theory to the study of the notion of singularity and the development of Laurent. Fourth, the application to the theory of residue and some of its practical consequences is another prupose of the course. In fifth and last place, it is about understanding and managing the tools of the infinite products and the main factorization theorems properly.