Unit COMPLEX ANALYSIS
- Course
- Mathematics
- Study-unit Code
- A001552
- Curriculum
- Generale
- Teacher
- Paola Rubbioni
- Teachers
-
- Paola Rubbioni
- Marco Cantarini (Codocenza)
- Hours
- 21 ore - Paola Rubbioni
- 21 ore (Codocenza) - Marco Cantarini
- CFU
- 6
- Course Regulation
- Coorte 2026
- Offered
- 2026/27
- Learning activities
- Caratterizzante
- Area
- Formazione matematica teorica avanzata
- Sector
- MATH-03/A
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Review of complex numbers. Elementary functions of a complex variable. Topology, limits, continuity, differentiability, sequences and series in Complex Analysis; the Cauchy–Riemann equations. Analytic functions; the theorems of Cauchy, Morera, and Goursat. Taylor and Laurent series; residue theory. Euler Gamma function. The Riemann zeta function and applications to analytic number theory.
- Reference texts
- 1. Carlo Presill, Elementi di Analisi Complessa, Volume 72, Springer, Second Edition, 2014.
2. Theodore Gamelin “Complex analysis” Springer Science Business Media, 2001.
3. Michael E. Taylor “Introduction to Complex Analysis”, Vol. 202, American Mathematical Society, 2020.
4. Course handouts.
5. Further didactical material available in Unistudium. - Educational objectives
- Acquisition of the basic knowledge of the theory of functions of a complex variable, aimed at a broader understanding of topics addressed in other courses and, more generally, at a deeper scientific maturity, necessary for approaching advanced texts. In particular, the study of Complex Analysis helps clarify certain aspects of the theory of functions of a real variable, such as the theory of line integrals and the domain of convergence of power series. The objective is to provide additional analytical tools and to develop problem-solving skills, including in applied contexts.
- Prerequisites
- Knowledge of the classical topics of Mathematical Analysis I and II, in particular the theory of real-valued functions of one and several variables, the theory of line integrals, and linear differential forms.
- Teaching methods
- The course is delivered through lectures, supported by software such as GeoGebra and NotebookLM. The theoretical material is complemented by exercises aimed at illustrating and further developing the topics covered.
- Other information
- The course is included in the General track of the Master’s Degree Programme in Mathematics; it is also recommended as an elective course within the Bachelor’s Degree Programme, in order to provide a stronger foundational preparation for students who intend to pursue a Master’s Degree in a track other than the General one.
- Learning verification modality
- The exam consists of an oral examination lasting approximately 30–45 minutes, accompanied by the solution of a few exercises. The purpose of the examination is to assess the level of understanding of the course topics, the ability to present them clearly and with awareness, and the acquired skills in solving simple exercises.
It is strongly recommended to take the examination after completing Mathematical Analysis II and III.
For information on support services for students with special educational needs and/or specific learning disorders, see: http://www.unipg.it/disabilita-e-dsa - Extended program
- Review of complex numbers; the Fundamental Theorem of Algebra. Functions of a complex variable; multivalued functions; the complex exponential and Euler’s formulas; trigonometric functions in the complex domain. Foundations of the theory of functions of one complex variable: elements of topology in Complex Analysis, limits and continuity; sequences and series in Complex Analysis, complex power series; differentiability of complex-valued functions of a complex variable; the Cauchy–Riemann equations. Analytic functions, line integrals, primitives of complex functions, Cauchy’s theorem, the Cauchy integral formula, the integral formula for derivatives, Morera’s theorem, Goursat’s theorem. Taylor and Laurent series; singularities and residue theory, with applications to integral calculus; Euler Gamma function: main properties. the Riemann zeta function: definition, functional equation, analytic continuation, Weierstrass product, zero-free region, Perron’s formula, and applications to analytic number theory.
- Obiettivi Agenda 2030 per lo sviluppo sostenibile
- Quality education