Complex Variable

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. (2 h)
6. Index of a point with respect to a closed path. (2 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 general methodological indications established in the Memory of the Degree in Mathematics of the USC will be followed.
In the lectures the essential contents of the subject will be presented.
In the interactive classes and tutorials, an active participation of the student will be sought and they may have room for different approaches in which concepts and issues of the subject are dealt with (problem solving, formalization of mathematical language, contrasting results obtained during the course, etc. )

The expository and interactive teaching will be face-to-face and will be complemented by a virtual course on the subject, in which students will find bibliographic materials, symbolic calculation programs for use on the Internet, exercise and test bulletins, podcast, or explanatory videos, etc. . Through the virtual course, students will also carry out exercises and assignments for continuous assessment, as described in the corresponding section.

The tutorials will be face-to-face or through telematic means (email, MS TEAMS official platform from USC, etc).

The general evaluation criteria established in the Memory of the Degree in Mathematics of the USC will be followed.

The continuous evaluation will foster the active participation in the classroom or virtual and it will mark the realization and resolution of problems or exercises commissioned by the teacher on practical or theoretical aspects of the subject, which may be individual or in groups.

With the different activities proposed, within the subject of Complex Variable and the fourth year of degree, the acquisition of competences, among others, CB2, CB3, CB4, CG2, CG3, CG4, CT1, CT2, CT3, CE7, CE8, CE9 as well as the capacity for teamwork and autonomous learning.
The exam will consist of theoretical and practical questions, and exercises and, in addition to the specific competences of the subject, the competencies CB1, CB2, CB4, CB5, CG1, CG3, CG4, CE1, CE2, CE3, CE4, CE5, will be evaluated. CE6.
The qualification obtained in the continuous evaluation will be valid in the two opportunities corresponding to the academic year.

The evaluation will be carried out by combining a continuous formative evaluation with a final test.

The continuous formative evaluation will consist of the collection of the tasks carried out in the classes throughout the course and a written test in the middle of the semester (if said test is agreed in the course coordination meetings).

The continuous assessment tasks will consist of carrying out standard exercises (including the use of MAPLE or a symbolic calculation program to which the student has access), writing demonstrations of theoretical results, tests in the virtual course, etc. All of them will be proposed in the same session in which they must be delivered, since they are not only assessment instruments, but mainly training exercises and reinforcement of the skills worked in the immediately preceding sessions. The teacher will comment on the tasks in the following sessions and each student will receive a grade between 0 and 10 points for each task.

Students who do not attend any of these sessions may present the same task before the next teaching hour of the same kind. In this case, unless there is a well-founded justification for the non-attendance, only 5 points can be achieved.

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 it were two tasks with the same mark.
The final test will be an exam in which the theoretical part of the subject will suppose, at least, 3 points of the 10 total.

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, 0.3 * C + 0.7 * F}

In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be an exam of the same type as the first.

The exam and the continuous assesment might not be exactly the same for both groups, but the tests taken by the students of different groups will be similar.

ON-SITE WORK AT CLASSROOM
Blackboard classes in large group (28 hours)
Seminaries (14 hours)
Laboratories (14 hours)
Tutorials (2 hours)
Evaluation activities (5 hours)
TOTAL: 63 hours
PERSONAL WORK OF THE STUDENT: 87

- 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); Cálculo Vectorial e Integración de Lebesgue (la parte de Cálculo Vectorial); Topología de los espacios euclidianos.

- Realizar las actividades que se propongan en las aulas.

- Estudiar con regularidad.

In cases of fraudulent performance of exercises or tests, it will be applied or included in the "Regulations for the validation of academic performance for students and for the review of qualifications".