Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 3 | 2 |
To be able to follow this course it is convenient to know the differential calculus in several variables.
To understand and know how to use the fundamental concepts and results in Complex Analysis.
To understand and know how to use the basic concepts of the Fourier transform.
Deep undersatnding of the proofs of the most important results and the most common techniques
1. Preliminaries. Complex numbers. Holomorphic functions and power series. Cauchy-Riemann equations.
2. Cauchy's Local Theory. Complex line integrals. Cauchy-Gourssat theorem and the Cauchy local theorem. Cauchy's integral formula. Holomorphy and analyticity. Analytical prolongation. Cauchy inequalities, Liouville's theorem and fundamental theorem of algebra. The maximun principle
3. The residue theorem. Laurent series and isolated singularities. Residue theorem and applications. The argument principle and the Rouché theorem .
4. Harmonic functions and Fourier Transform. Holomorphic functions and harmonic functions on a disk. Fourier transform.
Two weekly hours of Theory classes will be taught where the concepts will be scaled down and the important results (theorems) will be written to build the theory that we are introducing.
We will devote ourselves to demonstrating theorems and methods of resolution through examples and exercises.
The student will receive a list of exercises and problems about which we will work in the weekly class of problems. Previously, during your non-attendance activity, you will have read and thought the proposed exercises and problems. In this way, you can guarantee your participation in the classroom and facilitate the assimilation of the procedural contents.
The 3 seminar sessions will deal with complementary topics such as homographies; conformal representations; product of convolution and approximation of identity.
As a matter of course, students will have hours to ask questions to the teacher in the office of the professor.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Exercices | 14 | 0.56 | 10, 9, 3, 5, 6 |
Maths seminar | 6 | 0.24 | 10, 9, 3, 5, 6 |
Theory | 28 | 1.12 | 10, 9, 3, 5, 6 |
Type: Autonomous | |||
Studying time | 88 | 3.52 | 10, 9, 3, 5, 6 |
Learning mathematics is a complex process. It is a long-term process; In a sense, one can not appreciate the meaning of the first theorem until he has learned the last theorem.
There will be two written examinations during the semester, which will mainly consist of solving problems, but will also contain a theoretical part.
The student who is not present at 51% of the partial tests will have a "Not evaluable" final grade.
There will be two tests written during the semester (30% + 50%) and a seminar note (20%).
The final exam will be retrieved within the official exam period, which can also be chosen by the student who has passed to improve note.
The possible distinction will be awarded after completing the entire evaluation, possible recovery included.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First partial exam | 30 | 4 | 0.16 | 2, 4, 10, 1, 3, 8, 7 |
Maths Seminars | 20 | 2 | 0.08 | 10, 9, 3, 5, 6 |
Recovery exam | 80 | 4 | 0.16 | 2, 4, 10, 9, 1, 3, 5, 8, 7, 6 |
Second partial exam | 50 | 4 | 0.16 | 10, 3, 8, 7 |
L. Ahlfors; Complex Analysis. Mac-Graw Hill. Third edition, 1979.
J. Bruna and J. Cufí; Complex Analysis. EMS Textbooks in Mathematics, 2013.
W. Rudin, Real and Complex Analysis. Mac-Graw Hill, third edition, 1987.
E.M. Stein and R. Shakarchi; Complex Analysis. Princeton University Press, 2003.