Mathematical analysis
Code: 100094 ECTS Credits: 9| Degree | Type | Year | Semester |
|---|---|---|---|
| 2500149 Mathematics | OB | 2 | 1 |
Contact
- Name:
- Joaquim Bruna Floris
- Email:
- joaquim.bruna@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
Teachers
- Juan Jesús Donaire Benito
- Artur Nicolau Nos
Prerequisites
In order to be able, for a student, to follow the course, it is very important that the student has succeeded in the first course subject Funcions de Variable Real (functions of one real variable). If this is not the case, it is essential that, the student understands the notions of convergence of sequences and continuity, differentiability and integrability of functions. It is also crucial that the student has enough mathematical skills in the manipulation of limits, Taylor series representation of functions...
Objectives and Contextualisation
For a student to succeed in this subject is is essential to acquire the following capacities.
Theoretical skills.
1. Understand the notion of series convergence and improper integrals.
2. Know about the most important criteria to decide the convergence of series and improper integrals.
3.Fully understand the notion of uniform convergence of a sequence of functions.
4. Understand the results that relate the uniform convergence on one side, and the notions of continuity, derivability and integrability on the other.
5. Understand why it is important to consider power series in the complex context.
6. Understand the results that involve the regularity of functions defined from integrals depending on a parameter.
7. Know about the principal results that relate the regularity of a function and the convergence of a Fourier series.
8. Understand the utility of Fourier series.
9. Understand and be able to reproduce the proofs of the main results of the subject.
Problem solving skills
1. Be able to apply the different criteria to decide whether a series or an improper integral converge.
2. Be able to compute the radius of convergence of a power series and know how to sum them in some concrete situations.
3. Be able to represent a function as an infinite sum of terms, as a power series, if possible.
4. Prove results involving uniform convergence of sequences of functions.
5. Be able to compute the Fourier coefficients of functions and be able to compute the sum of some complex series applying the Fourier series results.
6. Be able to relate the different main results of the subject and apply them to solve concrete problems.
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.
- Demonstrate a high capacity for abstraction.
- 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.
- 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
- 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.
- Contrast acquired theoretical and practical knowledge.
- 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 the concepts of series and integral convergence and master the most important criteria of convergence.
- Understand the relationship between uniform convergence and continuity, derivability and integrability of the functions of a variable.
Content
1. Series of numbers.
1.1 Extension of the notion of limit of a sequence.
1.2 Notion of convergent series.
1.3 Non-negative series. Convergence criteria.
1.4 Absolute and condicional convergence.
1.5 Leibniz, Dirichlet and Abel criteria.
1.6 Rearranging series. The Riemann series theorem.
1.7 Infinite products.
2. Univorm convergence and power series.
2.1 Sequences of functions.
2.2 Pointwise and uniform convergence.
2.3 Uniform convergence and continuity, differentiability and integrability.
2.4 Function series.
2.5 Weierstrass M test.
2.6 Existence of continuous functions nowhere differentiable.
2.7 Power series and radius of convergence.
2.8 Abel Theorem.
2.9 Analytic functions.
2.10 Approximation of continuous functions by polynomials: Weierstrass theorem.
4. Improper Integrals.
4.1 Extension of the notion of Riemann integral for non-bounded functions or intervals.
4.2 Convergence of improper integrals.
4.3 Convergence criteria for positive functions.
4.5 Continuity and derivability for functions with more than one variable.
4.6 Integrals depending on one parameter.
4.7 The Euler Gamma function. Stirling's theorem.
5. Fourier series.
5.1 L^2 functions.
5.2 Trigonometric polynomials. Fourier coeficients. Fourier series.
5.3 Pointwise and uniform convergence of a Fourier series.
5.4 Gibbs phenomena.
5.5 Parseval's identity.
Methodology
It is just explained above.
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.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Final Exams | 4 | 0.16 | |
| Partial exams | 2 | 0.08 | |
| Problem sessions | 14 | 0.56 | |
| Seminar sessions | 14 | 0.56 | |
| Theory sessions | 42 | 1.68 | |
| Type: Supervised | |||
| Doubt clearing sessions student-professor | 4 | 0.16 | |
| Type: Autonomous | |||
| At home work | 46 | 1.84 | |
| Exam preparation | 30 | 1.2 | |
| Preparation | 4 | 0.16 | |
| Solve problems and exercises | 60 | 2.4 |
Assessment
Grading is based on four items:
a) Two mid-term exams, each corresponding to essentially one half of the syllabus, with grades P1,P2.
b) Submission of two set of exercisises through Campus Virtual, that might be revised individually. Their mean is LLEX. No resit possibility exists for this activity.
For those students having gone through these four tests, a mark C1 is generated according to C1=(0,4)*(P1+P2)+(0,2)*LLEX.
Next, a final exam with no previous requirements is held, with grade R, and a second mark C2 is generated according to C2=(0,8)*R+(0,2)*LLEX.
The final mark is max(C1,C2). Students with no C1,C2 grades will be considered as not assessable.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| First mid-term exam | 40% | 2 | 0.08 | 7, 5, 4 |
| Second mid-term exam | 40% | 2 | 0.08 | 2, 3, 7, 1, 6, 5, 4 |
| Submission of a set of exercises | 20% | 1 | 0.04 | 7, 6, 4 |
Bibliography
1. J. Casasayas i Mª C. Cascante. Problemas de análisis matemático. Edunsa Ediciones y Distribuciones Universitarias s.a., Barcelona, 1990.
2. F. Galindo i altres. Guía Práctica de Cálculo Infinitesimal en una variable real. Ed. Thomson, Madrid 2003.
3. J. M. Ortega. Introducció a l’Anàlisi Matemàtica. Manuals de la Universitat Autònoma de Barcelona 4, Bellaterra 1990.
4. C. Perelló. Càlcul Infinitesimal: amb mètodes i aplicacions. Enciclopèdia Catalana, Barcelona, 1994.
5. W. Rudin. Principios de Análisis Matemático. McGraw-Hill, Mèxic, 1981.
6. G. P. Tolstov. Fourier Series, Edover Publications, New York, 1976.
7. Laura Prat, Alejandro Molero, Apunts d'Anàlisi Matemàtica, available at Campus Virtual.
Software
None is needed