Program Overview

---

COMPLEMENTS OF COMPLEX ANALYSIS

Overview

The course introduces advanced topics in complex analysis.

Aims and Content

Learning Outcomes

The course covers advanced topics related to functions of a complex variable, including the Gamma function, Rouché's and Picard's theorems, the Riemann mapping theorem, and the main properties of functions of finite order. These topics allow students to develop a more mature and comprehensive understanding of complex analysis, preparing them to engage with more advanced problems in various areas of mathematics.

Aims and Learning Outcomes

The aim of the course is to introduce more advanced concepts of complex analysis, including in particular the theory of finite-order functions, biholomorphisms, and conformal mappings on subsets of C, as well as the theory of special functions. The course provides the necessary prerequisites to tackle more advanced topics in Analysis, Number Theory, Complex Geometry, and related subjects. By the end of the course, students are expected to be able to address advanced problems in complex analysis.

Prerequisites

The main prerequisite is the basic theory of complex analysis.

Teaching Methods

52 hours of frontal lectures.

Syllabus/Content

• Review of the basic theory of complex analysis
• Rouché's theorem
• Jensen's theorem
• Conformal maps
• Moebius' maps
• Riemann's mapping theorem
• Bloch and Picard theorems
• Theory of functions of finite order and theorems of Weierstrass and Hadamard
• a-points
• Special functions

Recommended Reading/Bibliography

• Notes on Aulaweb
• Remmert - Classical Topics in Complex Function Theory
• Stein, Shakarchi - Complex analysis
• Markushevich - Theory of Functions of a Complex Variable. Volume 2
• Viola - An introduction to special functions
• Tao - Online notes

Teachers and Exam Board

• Sandro Bettin
• Alessandro Fazzari

Exam Board

• Sandro Bettin (President)
• Alessandro Fazzari

Lessons

Lessons Start

February 26, 2026

Class Schedule

The timetable for this course is available.

Exams

Exam Description

Oral exam on the whole program.

Assessment Methods

During the oral exam, the student will have to be able to: state and prove the theorems presented in class, correctly state all the definitions, and solve simple exercises.

Exam Schedule

• Date: September 18, 2026
• Time: 09:00
• Location: GENOVA
• Degree type: Esame su appuntamento

Further Information

Students with disabilities or specific learning disorders (DSA) are reminded that, in order to request exam accommodations, they must first upload the relevant certification to the University website. The documentation will be verified by the University's Office for Inclusion Services for Students with Disabilities and SLD. Subsequently, well in advance (at least 7 days) before the exam date, students must send an email to the instructor responsible for the exam, including the course title, exam date, student's last name, first name, and student ID number, and the requested compensatory tools and dispensatory measures deemed necessary.

Agenda 2030 - Sustainable Development Goals

• Quality education
• Gender equality