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

LESSONS

LESSONS START

February 26, 2026

Class schedule

The timetable for this course is available here: Portale EasyAcademy

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. 

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 at servizionline.unige.it, in the “Students” section. 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, copying both the School's contact person for the inclusion of students with disabilities and DSA (sergio.didomizio@unige.it), and the above-mentioned University office. The email must include the following information:

Course title

Exam date

Student’s last name, first name, and student ID number

The requested compensatory tools and dispensatory measures deemed necessary

The inclusion contact person will confirm to the instructor that the student is entitled to request accommodations for the exam, and that such accommodations must be agreed upon with the instructor. The instructor will then inform the student whether the requested measures can be applied.

Requests must be sent at least 10 days before the exam date to allow the instructor sufficient time for evaluation. In particular, if concept maps are intended to be used during the exam (which must be much more concise than those used for study purposes), failing to respect the deadlines may make it impossible to guarantee enough time for any necessary adjustments.

For further information on requesting services and accommodations, please refer to the document “Guidelines for requesting services, compensatory tools and/or dispensatory measures, and specific aids.”