Logo UAB

Differential Equations and Complex Analysis

Code: 107611 ECTS Credits: 6
2025/2026
Degree Type Year
Physics OB 1

Contact

Name:
Ennio Salvioni
Email:
ennio.salvioni@uab.cat

Teachers

Fabrizio Rompineve Sorbello

Teaching groups languages

You can view this information at the end of this document.


Prerequisites

Previous knowledge of real-variable functions is required; therefore, it is advisable to have taken the course Calculus of One Variable.


Objectives and Contextualisation

The main objective of this course is to provide an introduction to the solution of ordinary differential equations and to the analysis of complex functions of a complex variable.


Learning Outcomes

  1. CM09 (Competence) Justify the use of calculus in one and several variables and differential equations in the resolution of general problems.
  2. CM10 (Competence) Adapt the basic mathematical strategy when approaching a given problem from an analytical point of view.
  3. KM10 (Knowledge) Describe the basic concepts of the calculation of several variables and the different methods of solving differential equations in their different typologies.
  4. KM11 (Knowledge) Identify the different types of differential equations and the essential concepts of complex variable analysis.
  5. KM11 (Knowledge) Identify the different types of differential equations and the essential concepts of complex variable analysis.
  6. SM07 (Skill) Apply the mathematical knowledge acquired to the resolution of mathematical and physical problems with mathematical representation.

Content

1) Introduction, equations and systems of differential equations, first-order equations. Exact differentials and integrating factor.

2) Second-order linear equations. Non-homogeneous equations. Equations with constant coefficients. Method of variation of parameters and undetermined coefficients. Higher-order linear equations.

3) Systems of linear ODEs. Homogeneous and non-homogeneous systems.

4) Complex numbers. Complex functions, differentiability. Cauchy-Riemann equations.

5) Cauchy's theorem and Cauchy's integral formula.

6) Series of functions of a complex variable. Laurent series. Isolated singularities.

7) Calculus of residues. Residue theorem. Application to the calculation of improper integrals.


Activities and Methodology

Title Hours ECTS Learning Outcomes
Type: Directed      
Exercise sessions 14 0.56 CM09, CM10, KM10, KM11, SM07, CM09
Seminars 8 0.32 CM09, CM10, KM10, KM11, SM07, CM09
Theory lectures 28 1.12 CM09, CM10, KM10, KM11, SM07, CM09
Type: Autonomous      
Problem solving 27 1.08 CM09, CM10, KM10, KM11, SM07, CM09
Study of theoretical concepts and methods 54 2.16 CM09, CM10, KM10, KM11, SM07, CM09

This course will consist of theory lectures, exercise sessions, and seminars:

  • Theory lectures: the definitions, theorems, and methods of the course are presented.
  • Exercise sessions: the solutions to some of the problems that will be assigned along the course are discussed.
  • Seminars: students solve problems in small groups in the classroom, under the supervision of a professor.

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
First partial exam 45% 3 0.12 CM09, CM10, KM10
Resit exam 90% 3 0.12 CM09, CM10, KM10, KM11, SM07
Second partial exam 45% 3 0.12 KM11, SM07
Seminars (2 submissions of problem set solutions) 10% 10 0.4 CM09, CM10, KM10, KM11, SM07

Continuous assessment
 
First partial exam: 45% of the final grade.

Second partial exam: 45% of the final grade.

Seminars (2 problem set submissions): 10% of the final grade. Four seminar sessions, each 2 hours long, are scheduled. Problems will be discussed in small groups of students under the supervision of a professor. After two of these sessions, students will have a fixed deadline to submit complete solutions of the problems covered in the seminar. Submissions may be prepared individually or in small groups of up to 5 people. The evaluation of each submission will represent 5% of the final grade.

The final grade will therefore be calculated as: 0.90 * (PartialExam1 + PartialExam2)/2 + 0.10 * (Submission1 + Submission2)/2.

To pass the course, the grade on each partial exam must be equal to or higher than 3 (out of 10) and the final grade for the course must be equal to or higher than 5.

Resit Exam:

The resit exam covers the entire program of the course.

To be eligible to take the resit exam, students must have been evaluated on both partial exams, with no minimum grade required.

Students may take the resit exam if they wish to improve their grade. In this case, the final grade for the exam portion will be the resit exam grade.

Single assessment  

The students that opted for single assessment evaluation will have to perform a final evaluation that will first consist of a test of the whole program of the course. This test will take place on the same date, time and place as the second partial exam of the continuous assessment modality. Besides, before the exam, the student will deliver 1 document with a solved problem set proposedat an earlier date. For the mark, 90% of the final mark will come from the exam and the problem set will count 10%. The students that opted for single assessement evaluation will have the chance of passing the module or improve their mark at the same re-evaluation test as the students that opted for the continuous assessement option (both exams will be identical and will take place on the same day, time and in the same place), but it is mandatory to at least have taken the previous final test. Only 90% of the grade can be improved at the re-evaluation test. The part related to the solved problem set cannot be improved.

Bibliography

Differential equations:

  • Elementary Differential Equations and Boundary Value Problems, W.E. Boyce, R.C. DiPrima & D.B. Meade, Wiley
  • Teoría y Problemas de Ecuaciones Diferenciales Modernas, R. Bronson, Compendios Schaum, McGraw-Hill

Complex variable:

  • Complex Variables, M. R. Spiegel, Schaum's Outline Series, McGraw-Hill
  • Càlcul en variable complexa, E. Bagan, A. Méndez i O. Pujolàs, Materials 243, Servei de Publicacions UAB

Software

There is no specific software for the course.


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
(PAUL) Classroom practices 1 Spanish second semester morning-mixed
(PAUL) Classroom practices 2 Spanish second semester afternoon
(SEM) Seminars 11 Spanish second semester morning-mixed
(SEM) Seminars 12 Spanish second semester afternoon
(TE) Theory 1 Spanish second semester morning-mixed
(TE) Theory 2 Spanish second semester afternoon