- Docente: Salvatore Coen
- Credits: 7
- SSD: MAT/03
- Language: Italian
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 8010)
Learning outcomes
Students have to master the fundamental concepts and methods of Complex Function Theory used in different branches of Mathematics. Emphasis to the geometrical point of view.
Course contents
Algebras of power series. Analytic functions. Exponential function. Elementary analytic functions. Argument, logarithm and general power functions . Linear differential forms and their integration. Winding number. Cauchy theory. Isolated singularities. The argument principle. Residue calculus. Harmonic functions. Other classical results. Applications.
Readings/Bibliography
H. Cartan "Théorie élémentaire des fonctions analytiques d' une ou plusieurs variables complexes", Hermann Ed. (an english translation is also available)
Teaching methods
Frontal lessons with exercises.
Assessment methods
The purpose of the exam is checking knowledge in the classical
theory of functions of a complex variable, with particular emphasis
on the geometric point of view of the theory. Students are
required to reconstruct the broad outline of the theorems, to
prove minor results and to solve minor problems. to solve practical
exercises (study of singularities, calculation of residues,
calculation of definite integrals by the residue calculus,
multilicative inverse of power series, reciprocal power
series, ..). to understand the main links with the programs of
which he has already taken the exam to provide at least a basic
historical framework of the theory.
=
Oral examination. Please note that this course forms a part of
an “integrated” course (“corso integrato”) that consists of two
different courses: Geometry 3 and Fundamentals of Advanced
Geometry. The examination of the "corso integrate" can be taken in
two ways The student may take a single examination (on the program
of both of the courses). The student may take the examination
in two different exam sessions. In the latter case, the
student takes an examen in one of the two courses (in his
preference; this is a mid-term exam) and he will be evaluated
for this examination. Once his examination result is accepted
(grade ≥ 18) it will no longer be possible to give other tests on
the same course. In a second time, the student has to
take the exam of the other course and he will be
evaluated for this test (mark ≥ 18) The final
assessment is decided on the basis of the evaluations of both of
courses.
Teaching tools
Blackboard.
Office hours
See the website of Salvatore Coen