Mathematical Analysis
Code: 100094Credits: 9
| Degree programme | Type | Course |
|---|---|---|
| Mathematics | OB | 2 |
Contact lecturer
- Name :
- Artur Nicolau Nos
- Email :
- artur.nicolau@uab.cat
Teaching staff
- Artur Nicolau Nos
- Joan Hernandez Garcia
- Laura Prat Baiget
Group languages
You can consult this information at the end of the document.
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
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.
Learning outcomes
- CM16 (Construct rigorous proofs of intermediate results in mathematical analysis, multivariable calculus, and complex analysis.) Construct rigorous proofs of intermediate results in mathematical analysis, multivariable calculus, and complex analysis.
- KM26 (Define the concepts of series and integral convergence, with the aim of mastering the most important convergence criteria.) Define the concepts of series and integral convergence, with the aim of mastering the most important convergence criteria.
- SM22 (Interrelate the concepts of uniform convergence, continuity, differentiability and integrability of functions of one or more real variables.) Interrelate the concepts of uniform convergence, continuity, differentiability and integrability of functions of one or more real variables.
Contents
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.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Exam preparation | 30 | 1.2 | |
| Preparation | 4 | 0.16 | |
| At home work | 46 | 1.84 | |
| Partial exams | 2 | 0.08 | |
| Seminar sessions | 14 | 0.56 | |
| Solve problems and exercises | 60 | 2.4 | |
| Doubt clearing sessions student-professor | 4 | 0.16 | |
| Theory sessions | 42 | 1.68 | |
| Final Exams | 4 | 0.16 | |
| Problem sessions | 14 | 0.56 |
It is just explained above.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Seminars | 10% | 1 | 0.04 | CM16, KM26, SM22 |
| Second mid-term exam | 45% | 2 | 0.08 | CM16, KM26, SM22 |
| First mid-term exam | 45% | 2 | 0.08 | CM16, KM26, SM22 |
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)Two Seminars. 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=(0,45)*(P1+P2)+(0,1)*LLEX.
Next, a final exam for students with C1<5 with grade R, and a second mark C2 is generated according to C2=min{(0,9)*R+(0,1)*LLEX, 5}.
The final mark is max(C1,C2). Students with no C1,C2 grades will be considered as not assessable.
Unique evaluation
Those students having chosen «unique evaluation» will be required, the day the second partial test takes place, to:
- The grade LLEX will be decided upon a personal interview.
- Take a final exam, with grade F. The grade will be C1=(0,1)*LLEX+(0,9)*F.
In case C1<5, students may take a recovery exam, with grade R, to be held at a posterior date decided by the degree coordinator. The final grade will be C2=(0,1)*LLEX+(0,9)*R
Bibliography
M.Spivak. Calculus. Càlcul Infinitesimal. Ed. Reverté, 1995
- W. Rudin. Principles of Mathematical Analysis. Third Edition. McGraw-Hill 1976
- R. Bartle, D.Sherbert, Introducción al análisis Matemático de una variable, Limusa-Willey, 2010
- T. Körner, Fourier Analysis, Cambridge University Press, 1988
- J. Stillwell, Mathematics and its History, Springer, 2012
Software
None is needed
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | first semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | first semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 2 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 3 | Catalan | first semester | morning-mixed |