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2020/2021

Complex and Fourier analysis

Code: 100103 ECTS Credits: 6
Degree Type Year Semester
2500149 Mathematics OB 3 2
The proposed teaching and assessment methodology that appear in the guide may be subject to changes as a result of the restrictions to face-to-face class attendance imposed by the health authorities.

Contact

Name:
Juan Eugenio Mateu Bennassar
Email:
Joan.Mateu@uab.cat

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
No
Some groups entirely in Catalan:
Yes
Some groups entirely in Spanish:
No

Prerequisites

To be able to follow this course it is convenient to know the differential calculus in several variables.

Objectives and Contextualisation

 

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

Competences

  • Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  • Apply critical spirit and thoroughness to validate or reject both one’s own arguments and those of others.
  • Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
  • Calculate and reproduce certain mathematical routines and processes with agility.
  • Recognise the presence of Mathematics in other disciplines.
  • Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  • Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  • Understand and use mathematical language.
  • Use computer applications for statistical analysis, numeric and symbolic calculus, graphic display, optimisation or other purposes to experiment with Mathematics and solve problems.

Learning Outcomes

  1. Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  2. Apply critical spirit and thoroughness to validate or reject both one’s own arguments and those of others.
  3. Comfortably deal with the calculation of residues and its applications
  4. Contrast acquired theoretical and practical knowledge.
  5. Handle with ease homographic transformations and conformal representation.
  6. Know how to calculate Fourier periodic function coefficients and their possible immediate applications for calculating sums of series.
  7. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  8. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  9. Understand Fourier and Laplace transforms of elemental functions and their application to the resolution of differential equations.
  10. Understand the basic results and fundamental properties of holomorphic functions and Cauchy theory.

Content

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.

Methodology

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.

Activities

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

Assessment

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.

Assessment Activities

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

Bibliography

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.