Complex Fourier Analysis
Code: 104400 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Computational Mathematics and Data Analytics | OB | 2 |
Contact
- Name:
- Martí Prats Soler
- Email:
- marti.prats@uab.cat
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
Elementary Algebra and differential and integral Calculus.
Objectives and Contextualisation
- Understand and use the concepts and fundamental results of Complex Analysis.
- Understand and use the basic concepts of the Fourier series and the Fourier transform.
- Apply the results of this area in various situations: circuits, fluid theory, signal processing, resolution of differential equations, etc.
Learning Outcomes
- CM20 (Competence) Calculate Fourier coefficients of periodic functions and their possible immediate applications to the calculation of sums of series.
- CM21 (Competence) Select the suitable data compression in each case to preserve the desired properties.
- KM16 (Knowledge) Identify the basic results and the fundamental properties of holomorphic functions, Cauchy's theorem, the Fourier and Laplace transforms of elementary functions, and their application to the solution of differential equations.
- KM17 (Knowledge) Identify the relationship between uniform convergence and continuity, derivability or integrability of functions of one variable. (
Content
- Complex numbers. Analytic functions. Power series.
- Cauchy local theory.
- Residues.
- Fourier series.
- Harmonic functions and Fourier transform.
- Applications.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Lectures | 30 | 1.2 | |
| Problem session | 12 | 0.48 | |
| Working seminars | 11 | 0.44 | |
| Type: Autonomous | |||
| Solving problems | 58 | 2.32 | |
| Studying theoretical concepts | 30 | 1.2 |
See the catalan version
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.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Final exam | 40% | 3.6 | 0.14 | CM20, CM21 |
| Midterm exam | 40% | 3.6 | 0.14 | KM16, KM17 |
| Submission of exercise sets | 20% | 1.8 | 0.07 | CM20, CM21, KM16, KM17 |
If the teaching staff deems it appropriate, interviews with students may be requested to adjust the final grades. Throughout the course, other activities that may contribute to improving the final grade can be offered, such as participation in forums, individual tasks, or projects. These grade improvements will only apply if the student achieves a partial exam average above 3.75.
For example, if a project is proposed with a 10% weight, the continuous assessment grade will be calculated as:
QC = 0.9·QP + 0.1·max(QP, T),
where T is the grade for the project.
If QC ≥ 5, the course is considered passed.
Otherwise, the student may take a resit exam, obtaining grades R1 and R2 corresponding to the recovery of each partial exam. Then, the resit grade will be:
R = (max(P1, R1) + max(P2, R2)) / 2,
and the final grade will be:
QR = min(0.8·R + 0.2·S, 5),
which means that the maximum grade attainable in the resit is a 5.
The final course grade will always be:
QF = max{QC, QR}.
Possible distinctions (honours) will be awarded in accordance with current regulations, once the entire evaluation process is complete.
If a student has taken only one assessment test, the final mark will be recorded as “Not evaluable”.
Unique assessment
Those students pledging for unique assessment, will have to solve a final test versing about all the content of the subject.
The final mark will be obtained by a mean of the submission of exercise sets (20%) and the final test (80%).
In case the mark is below 5, the student will have a second chance in the recovery test. Its date will be fixed by the coordination of the degree. In this test the student may recover the 80% corresponding to the tests. The submission part will not be reevaluated.
Bibliography
Bibliografia bàsica
-
C. Cascante, N. Fagella, E. Gallego, J. Pau i M. Prats, Apunts d'Anàlisi Complexa. Versió preliminar disponible en línia.
-
L. Ahlfors, Complex Analysis, McGraw-Hill, 3a edició, 1979.
(Referència clàssica que, amb un format compacte, tracta molts temes amb gran rigor.) -
J. Conway, Functions of One Complex Variable, 2a edició, Springer-Verlag, 1978.
(Abarca molt més que el curs i inclou nombrosos problemes.) -
J. P. D'Angelo, An Introduction to Complex Analysis and Geometry, AMS, 2010.
(Introducció de nivell més elemental que les obres anteriors.) -
B. Davis, Transforms and Their Applications, 3a edició, Springer, 2001.
(Serveix com a inici i aprofundimenten l’estudi de les transformacions integrals.) -
M. C. Pereyra i L. A. Ward, Harmonic Analysis: From Fourier to Wavelets, AMS, 2012.
(Curs força complet d’anàlisi harmònica.)
Bibliografia complementària
- J. Bruna i J. Cufí, Anàlisi Complexa, Manuals UAB 49, 2008.
-
L. Volkovyski, G. Lunts i I. Aramanovich, Problemas sobre la teoría de funciones de variable compleja, MIR, 1977.
-
R. Burckel, Introduction to Classical Complex Analysis, vol. I, Academic Press, 1979.
-
W. Rudin, Análisis Real y Complejo, Alhambra, 1979.
-
S. Saks i A. Zygmund, Fonctions Analytiques, Masson et Cie, 1970.
-
E. Stein i R. Shakarchi, Complex Analysis, Princeton University Press, 2003.
-
R. N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill, 1986.
-
R. M. Gray i J. W. Goodman, Fourier Transforms, Kluwer, 1995.
-
R. V. Churchill i J. W. Brown, Complex Variables and Applications, 2009.
Software
- Sagemath: https://www.sagemath.org
- Maxima: https://maxima.sourceforge.io
- WxMaxima: https://wxmaxima-developers.github.io/wxmaxima/index.html
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PLAB) Practical laboratories | 1 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | second semester | morning-mixed |
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |