ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
To provide the student, in their first contact with the theory of functions of complex variable, tools, techniques and basic concepts of this theory, by highlighting main properties of the complex analysis, their differences with the real analysis studied in previous years and making clear the application of the aforementioned theory to solve some problems of real analysis.
COMPLEX DIFFERENTIABILITY
1. Complex numbers. The Euclidean plane and the complex plane. (1 h)
2. The extended complex plane and the Riemann sphere: the point at infinite. Qubits. (1 h)
3. Complex differentiability. Cauchy-Riemann equations. Holomorphic functions. (2 h)
4. Elementary functions of a complex variable. (2 h)
CAUCHY INTEGRAL THEOREM
5. Integration throughout a path. (1 h)
6. Index of a point with respect to a closed path. (1 h)
7. Local version of the Cauchy integral theorem: local primitive. (2 h)
8. Analyticity of the holomorphic functions. Morera's theorem. (2 h)
9. Zeros of holomorphic functions: uniqueness theorem. (1 h)
10. Entire function. (2 h)
11. Liouville's theorem. Fundamental theorem of algebra. (1 h)
12. Cauchy’s integral theorem. (2 h)
13. Harmonic functions. (1 h)
ISOLATED SINGULARITY
14. Laurent’s series. (2 h)
15. Isolated singularity: Casorati-Weierstrass theorem. (2 h)
16. Residues. Residue theorem. Applications. (2 h)
17. Zeta Riemann function. (1 h)
Basic:
JAMESON, G. J. O.: A First Course on Complex Functions. Chapman and Hall. 1982.
MÁRQUEZ, I., NIETO, J.J.: Variable Compleja, NINO-CID, 2017.
Additional:
APOSTOL, T.M.: Análisis Matemático. Reverté, 1986.
CONWAY, J. B.: Functions of One Complex Variable I. Springer. 1978.
GÓMEZ LÓPEZ, M. - CORDERO GRACÍA, M.: Variable compleja. 50 problemas útiles. García-Maroto editores, S.L. 2007
LOPEZ-GOMEZ, J.: Ecuaciones Diferenciales y Variable Compleja. Prentice Hall, , 2001.
In this course, we aim at preparing the student to acquire basic competences as indicated in the Memory of the Mathematics Degree at the USC: CB1, CB2, CB3, CB4, CB5; general CG1, CG2, CG3, CG4, CG5; transversal competences CT1, CT2, CT3, CT5; and specific CE1, CE2, CE3, CE4, CE5, CE6, CE7, CE8 and CE9.
To understand and use the basic concepts of functions of one complex variable.
To know the relation between the holomorphic and the analytic functions.
To calculate residues and to make use of them for the determination of real integrals.
The teaching is programmed in classes in large group and in reduced group and tutorials in very reduced groups or individualized.
In blackboard classes will be presented part of the contents of the subject; will be given suggestions for complete the contents and will be proposed and solved problems or exercises.
The tutorials will be, firstly, to make clear doubts about theory, problems, exercises or to propose other tasks and also will be a means in order that the students propose or refute in a reasoned way the arguments or ideas to be exposed there that will take as reference for the continuous assessment of the subject. It is necessary, therefore, not only the presence but also the active participation of the student body in this type of activities designed to facilitate the process of apprenticeship of the subject on a solid basis.
The mark of each student will be made by means of two instruments, continuous assessment (with a weight of 30%) and realization of a theoretic-practical final probe, which will take place according to the exam calendar fixed by the Faculty of Mathematics.
The mark of the continuous formative evaluation will be the average of the marks of the tasks, including the mark of the written test with double weight, that is, as if they were two tasks with the same mark.
The final test will be an exam in which the theoretical part of the subject will involve, at least, 3 points out of the 10 totals.
With the mark of the continuous formative assessment (C) and the mark of the final test (F) the final mark in the subject (NF) will be calculated according to the following formula:
NF = max {F, 0.65 * C + 0.35 * F}
NOTE. It is possible to pass the subject without taking the final test (previous formula with F = 0). It is understood as NOT PRESENTED who at the end of the teaching period is not in a position to pass the subject without taking the final test and does not appear at said test.
In any case, we will adhere, depending on the different scenarios, to the document “Directrices para o desenvolvemento dunha docencia presencial segura no curso 2020/21”.
With the activities proposed, and within the subject of Complex Variable and the frame of the fourth year of the Degree, it will be assessed the acquisition of competences, among others, CB2, CB3, CB4, CG2, CG3, CG4, CT1, CT2, CT3, CE7, CE8, CE9 or the ability to work in group and learn independently.
The exam will consist of theoretical and practical questions, and, in addition to the specific competences of the subject, it will evaluate the competences CB1, CB2, CB4, CB5, CG1, CG3, CG4, CE1, CE2, CE3, CE4, CE5, CE6.
The grade obtained in the continuous assessment is valid in both opportunities of the academic year.
In cases of fraudulent performance of exercises or exams, it will be applied or included in the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións".
ON-SITE WORK AT CLASSROOM
Blackboard classes in large group (26 hours)
Seminaries (13 hours)
Laboratories (13 hours)
Tutorials (2 hours)
Evaluation activities (5 hours)
TOTAL: 59 hours
PERSONAL WORK OF THE STUDENT: 91
- Tener cursadas las siguientes asignaturas : Introducción al análisis matemático; Continuidad y derivabilidad de funciones de una variable real; Integración de funciones de una variable real; Diferenciación de funciones de varias variables reales; Series funcionales e integración de Riemann de varias variables reales (excepto la parte correspondiente a integración de varias variables reales); Topología de los espacios euclidianos.
- Realizar las actividades que se propongan en las aulas.
- Estudiar con regularidad.
In any case, we will adhere, depending on the different scenarios, to the document “Directrices para o desenvolvemento dunha docencia presencial segura no curso 2020/21”.
In cases of fraudulent performance of exercises or exams, it will be applied or included in the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións".
Juan José Nieto Roig
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813177
- juanjose.nieto.roig [at] usc.es
- Category
- Professor: University Professor
José Manuel Uzal Couselo
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- josemanuel.uzal [at] rai.usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 15:00-16:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| Tuesday | |||
| 15:00-16:00 | Grupo /CLIS_03 | Spanish | Ramón María Aller Ulloa Main Hall |
| 16:00-17:00 | Grupo /CLIS_02 | Spanish | Ramón María Aller Ulloa Main Hall |
| 17:00-18:00 | Grupo /CLIS_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Wednesday | |||
| 15:00-16:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 17:00-18:00 | Grupo /CLIL_01 | Spanish | Computer room 2 |
| 18:00-19:00 | Grupo /CLIL_02 | Spanish | Computer room 2 |
| Thursday | |||
| 15:00-16:00 | Grupo /CLIL_05 | Spanish | Computer room 3 |
| 16:00-17:00 | Grupo /CLIL_03 | Spanish | Computer room 3 |
| 18:00-19:00 | Grupo /CLIL_04 | Spanish | Computer room 3 |
| 01.22.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 02 |
| 01.22.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 03 |
| 01.22.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 01.22.2021 16:00-20:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 06.28.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |