ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Center Faculty of Mathematics
Call: First Semester
Teaching: Sin Docencia (En Extinción)
Enrolment: No Matriculable (Sólo Planes en Extinción)
To introduce the student, with the support of examples and practice, in the comprehension of the first structure of the Mathematical Analysis: the ordered and complete field of the real numbers.
To introduce and consolidate, with examples and exercises, the notions of convergence of sequences and of numerical series.
To present, practising with the different notations, the operations with complex numbers.
1. REAL NUMBERS
1.1 Natural numbers. Principle of induction.
1.2 Rational numbers. Countability.
1.3 Axiomatic of the real numbers (R). Supreme axiom and consequences.
1.4 Archimedian property of R. Density of Q in R. Topology of the real line.
2. SEQUENCES OF REAL NUMBERS
2.1 Intuitive introduction to the concepts of sequence and limit. Generalities.
2.2 Convergent sequences and their limits. Properties.
2.3 Infinite limits.
2.4 Convergence and divergence of monotone sequences.
2.5 Subsequences. Bolzano-Weierstrass Theorem. Limits of oscillation.
2.6 Cauchy sequences. Completeness of R.
2.7 Calculus of limits. Stirling and Stolz criteria.
3. SERIES OF REAL NUMBERS
3.1 Intuitive introduction to the concepts of series and their sum.
3.2 Numerical series. Convergence of series.
3.3 Series of non-negative terms. Convergence criteria.
3.4 Absolute and conditional convergence. Criteria of non-absolute convergence.
3.5 Real decimal expansion and other numerical systems.
4. COMPLEX NUMBERS
4.1 Complex numbers. Expressions, operations and roots of complex numbers.
4.2 Exponential form and its consequences: powers, roots and Euler and de Moivre formulae.
BASIC:
[1] T.M. Apostol. Análisis Matemático (2ª Ed.). Reverté, 1979.
[2] R.G. Bartle and D.R. Sherbert. Introducción al Análisis Matemático de una Variable (3ª Ed.). Limusa Wiley, 2010.
[3] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Números reais. Universidade de Santiago de Compostela, 2022.
[4] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Números complexos. Universidade de Santiago de Compostela, 2022.
[5] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Series de números reais. Universidade de Santiago de Compostela, 2022.
[6] R. Figueroa Sestelo and Ó.A. Otero Zarraquiños. Sucesións de números reais. Universidade de Santiago de Compostela, 2022.
COMPLEMENTARY:
[1] S. Behar Jequín, R. Roldán Inguanzo and A. Arredondo Soto. Análisis matemático real: ejercicios y problemas. Universidad de La Habana, 2021.
https://elibro-net.ezbusc.usc.gal/es/lc/busc/titulos/196988
[2] J. Casasayas and M.C. Cascante. Problemas de Análisis Matemático de una variable real. Edunsa, 1990.
[3] A. García López et al. Cálculo I. Teoría y problemas de Análisis Matemático en una variable (2ª Ed.). Clagsa, 1994.
[4] R. Magnus. Fundamental Mathematical Analysis. Springer Cham, 2020.
https://link.springer.com/book/10.1007/978-3-030-46321-2
[5] T. Radozycki. Solving problems in Mathematical Analysis, Part I. Springer Nature, 2020.
https://link.springer.com/book/10.1007/978-3-030-35844-0
[6] B.S.W. Schröder. Mathematical Analysis: A Concise Introduction, John Wiley & Sons, 2007.
https://onlinelibrary.wiley.com/doi/book/10.1002/9780470226773
[7] M. Spivak. Calculus (2ª Ed.). Reverté, 1994.
In addition to contribute to achieve the general and transverse competences taken up in the memory of the degree, this subject will allow the student to get the following specific competences:
CE1 - To understand and use mathematical language;
CE2 - To know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstrac properties and material facts of a problem, distinguishing them from those purely occasional or incidental.
The assessment will be carried out by a final exam.
TRABAJO PERSONAL DEL ALUMNO
Estudio autónomo individual o en grupo (57 horas)
Escritura de ejercicios, conclusiones u otros trabajos (20 horas)
Programación/experimentación u otros trabajos en ordenador/laboratorio (10 horas)
Lecturas recomendadas, actividades en biblioteca o similar (5 horas)
Total de horas de trabajo personal del alumno: 92
To study every day by using the bibliographical material. To read carefully the theoretical part until it has been assimilated and then, to answer the corresponding questions, exercises or problems.
Rodrigo Lopez Pouso
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813166
- rodrigo.lopez [at] usc.es
- Category
- Professor: University Professor
Érika Diz Pita
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813202
- erikadiz.pita [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| 01.23.2026 16:00-20:00 | Grupo de examen | Classroom 06 |
| 06.11.2026 16:00-20:00 | Grupo de examen | Classroom 06 |