Differential Equations and Complex Analysis
Code: 107611Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Physics | OB | 1 |
Contact lecturer
- Name :
- Ennio Salvioni
- Email :
- ennio.salvioni@uab.cat
Teaching staff
- Fabrizio Rompineve Sorbello
Group languages
You can consult this information at the end of the document.
Prerequisites
Previous knowledge of real-variable functions is required; therefore, it is advisable to have taken the course Calculus of One Variable.
Objectives
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
- CM09 (Justify the use of calculus in one and several variables and differential equations in the resolution of general problems.) Justify the use of calculus in one and several variables and differential equations in the resolution of general problems.
- CM10 (Adapt the basic mathematical strategy when approaching a given problem from an analytical point of view.) Adapt the basic mathematical strategy when approaching a given problem from an analytical point of view.
- KM10 (Describe the basic concepts of the calculation of several variables and the different methods of solving differential equations in their different typologies.) Describe the basic concepts of the calculation of several variables and the different methods of solving differential equations in their different typologies.
- KM11 (Identify the different types of differential equations and the essential concepts of complex variable analysis.) Identify the different types of differential equations and the essential concepts of complex variable analysis.
- SM07 (Apply the mathematical knowledge acquired to the resolution of mathematical and physical problems with mathematical representation.) Apply the mathematical knowledge acquired to the resolution of mathematical and physical problems with mathematical representation.
Contents
1) Introduction, differential equations and systems of differential equations, first-order equations. Exact differentials and integrating factor.
2) Second-order linear equations. Homogeneous and non-homogeneous equations. Equations with constant coefficients.
3) Method of variation of parameters and undetermined coefficients. Higher-order linear equations.
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.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Exercise sessions | 14 | 0.56 | CM09, CM10, KM10, KM11, SM07 |
| Seminars | 8 | 0.32 | CM09, CM10, KM10, KM11, SM07 |
| Theory lectures | 28 | 1.12 | CM09, CM10, KM10, KM11, SM07 |
| Problem solving | 29 | 1.16 | CM09, CM10, KM10, KM11, SM07 |
| Study of theoretical concepts and methods | 58 | 2.32 | CM09, CM10, KM10, KM11, SM07 |
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.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Seminars (2 submissions of problem set solutions) | 10% | 4 | 0.16 | CM09, CM10, KM10, KM11, SM07 |
| Second partial exam | 45% | 3 | 0.12 | KM11, SM07 |
| Resit exam | 90% | 3 | 0.12 | CM09, CM10, KM10, KM11, SM07 |
| First partial exam | 45% | 3 | 0.12 | CM09, CM10, KM10 |
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 are scheduled, each lasting 2 hours, where problems will be discussed in small groups of students (maximum 5 people) under the supervision of a professor. During two of these sessions, with mandatory presence, students will work on the assigned problems in small groups and with the supervision of professors, and will submit the completed work at the end of the seminar session. 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 the solutions to one problem set proposed at 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 resit exam as the students that opted for the continuous assessment 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.
Not assessable
Students who are not assessed in more than 50% of the assessable activities of the course, will receive the grade of not assessable.
Use of AI
In this course, the use of Artificial Intelligence (AI) technologies is not permitted in any assessed activity. Any assessable work that includes fragments generated with AI will be considered a lack of academic honesty and may lead to a partial or total penalty in the grade of the activity, or greater sanctions in serious cases.
Bibliography
Differential equations:
- Ordinary Differential Equations, M. Tenenbaum & H. Pollard, Dover
- Elementary Differential Equations and Boundary Value Problems, W.E. Boyce, R.C. DiPrima & D.B. Meade, Wiley
Complex variable:
- Càlcul en variable complexa, E. Bagan, A. Méndez & O. Pujolàs, Materials 243, Servei de Publicacions UAB
- Mètodes matemàtics. Variable complexa, J. Peñarrocha, A. Santamaria & J. Vidal, Publicacions Universitat València
Software
There is no specific software for the course.
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 | Spanish | second semester | morning-mixed |
| (PAUL) Classroom practices | 1 | Spanish | second semester | morning-mixed |
| (TE) Theory | 2 | Spanish | second semester | afternoon |
| (PAUL) Classroom practices | 2 | Catalan/Spanish | second semester | afternoon |
| (SEM) Seminars | 11 | Catalan/Spanish | second semester | morning-mixed |
| (SEM) Seminars | 12 | Catalan/Spanish | second semester | morning-mixed |
| (SEM) Seminars | 21 | Catalan/Spanish | second semester | afternoon |
| (SEM) Seminars | 22 | Catalan/Spanish | second semester | afternoon |