Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OB | 3 | 2 |
It is a third-year subject, so students already have some mathematical background needed to follow it. Although it will be quite self-contained some prior knowledge is essential. For example, series theory and power series in Anàlisi Matemàtica and differential calculus in Càlcul en Diverses Variables.
Know and be able to use the fundamental concepts and results of Complex Analysis.
Understand in depth the demonstrations of the most important results and the most common techniques in the area.
Have some initial notions of the basic concepts of the Fourier transform and the Laplace transform.
1. Preliminaries. Complex numbers. Power series. Holomorphic functions. Cauchy-Riemann equations.
2. Cauchy's Local Theory. Complex line integrals. Cauchy-Goursat theorem and the local Cauchy theorem. Holomorphy and analyticity. Zeros of holomorphic functions. The index of a closed curve. Cauchy's integral formula. Analytical extension. Cauchy inequalities, Liouville's theorem and Fundamental theorem of algebra. The principle of the maximum module. Schwarz's motto.
3. Singularities. Laurent series. Classification of isolated singularities. Waste theorem and applications. The principle of argument and Rouché's theorem.
4. Harmonic functions. Basic properties of harmonic functions. Harmonic functions on a disc. Dirichlet problem.
5. Integral transforms. Fourier transform. Laplace transform. Basic properties. Applications to solve equations.
The subject has two hours of theory every week. They will be taught traditionally with blackboard. In the theory where the concepts will be shelled and enunciating the important results (theorems) that build the theory that we are introducing.
We will dedicate ourselves to proving theorems and methods of solving by means of examples and exercises.
The student will receive a list of exercises and problems that we will work on in the weekly problem class. Previously, during their house activity, they will have read and thought about the proposed exercises and problems. In this way, their participation in the classroom can be guaranteed and the assimilation of the procedural contents will be facilitated.
There will be three seminar sessions, each of them of two hours. In the first two sessions there will be a first part where the teacher will complement two topics already covered in the theory and problem classes. In the second part, the students will do independently some problem related to what has been explained, it can be done in groups.
The third session of the seminars will be evaluable. If the sanitary conditions allow it will be done in pairs. The topics covered are a more in-depth study of Möbius transformations and further applications of the residue theorem in the calculation of definite integrals. The evaluation will deal with these issues.
In the event that we are forced to teach electronically, sufficient material will be provided for monitoring. The Virtual Campus will be the means of communication between teachers and students. It will be important to consult it on a daily basis.
Students will have a tutoring and advice service both online and in the office. It is recommended to use this help for the follow-up of the course.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
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. Maturation is needed that is achieved throughout the course. Many times, some result from the beginning of the theory is understood to be completely advanced in the course. This shows the difficulty of the evaluations.
In the university there is the model of continuous evaluation which is not viable as it is done in secondary education as there is neither the logistics nor the possibilities to carry it out. Then a model is made, which has a certain resemblance to a continuous evaluation, and which forces the students to do the study we can say every day.
There will be two written partial exams during the semester, which will mainly consist of problem solving, but will also contain a theoretical part. They will have a P1 and P2 grade respectively. The first will have a weighting of 35% and the second 45%.
The seminar test will assign an S grade of up to 20%.
The qualification for continuous assessment will be obtained with the formula
QC = 0.35 * P1 + 0.45 * P2 + 0.2 * S.
If QC is greater than or equal to 5 the course will be passed. Otherwise, the student will be able to present himself to a recovery, and will obtain a grade R and one has
QC’= 0.8 * R + 0.2 * S.
In order to appear for recovery, the minimum of P1 and P2 is required to be greater than or equal to 0.5. You can also choose to apply for people who want to improve their grade. The course note will always be
QF=maximum {QC, QC’}.
Possible honors registrations will be awarded in compliance with current regulations and once the entire assessment has been completed, possible recovery included.
If a student has only taken an assessment avaluation they will be given a "No avaluable" final grade.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First partial exam | 35 | 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 | 45 | 4 | 0.16 | 10, 9, 3, 8, 7 |
Bibliografia bàsica:
1) L. Ahlfors, Complex Analysis. Mc Graw-Hill. 3ra edició, 1979.(És una referència clàssica que amb un format reduït tracta moltíssims temes de forma rigorosa).
2) J. Conway, Functions of One Complex Variable, second Edition, Springer Verlag, 1978. (Abarca molt més que el curs i conté molts problemes).
3) J. P. D'Angelo; An introduction to Complex Analysis and Geometry; A.M.S. 2010 (És una introducció de nivell molt més elemental que els anteriors).
4) B. Davis; Transforms and Their Applications, Thrid Edition, Springer (2001) (Serveix com a inici i aprofundiment en l’estudi del món de les transformacions integrals).
Bibliografia complementària:
1) J. Bruna, J. Cufí, Anàlisi Complexa, Manuals UAB 49, 2008.
2) R. Burckel, Introduction to classical complex Analysis, vol I, Academic Pres, 1979.
3) W. Rudin, Análisi Real y Complexo, Alhambra, 1979
4) S. Saks et A. Zygumund, Fonctions Analytiques, Massin et Cie, 1970.
5) M. Stein, R: Shakarchi, Complex Analysis, Princeton University Press, 2003.
There are no computer practice classes in the subject, so no study of computer programs will be done. Despite this, it will be recommended to use mathematical manipulation programs such as Maxima or Wolfram Alpha, which can be very useful.