Degree | Type | Year | Semester |
---|---|---|---|
2500097 Physics | OB | 2 | 2 |
Prior knowledge of real variable functions is required, so it is advisable to have studied the Calculus I, Calculus II and Calculus of Several Variables.
The main goal of this course is to introduce the analysis of complex functions of a complex variable, its calculation and applications, beginning with the presentation of complex numbers and ending with advanced applications and topics.
1) Complex numbers: representation, Euler's formula, powers and roots
2) Topology of complex numbers
3) Elementary and multiple-valued functions: exponential, trigonometric, hyperbolic, logarithm, power
4) Fourier series and transform
5) Complex differentiation: limits and continuity, Cauchy-Riemann equations, differentiation
6) Cauchy's theorem: integrals in the complex plane, primitives
7) Cauchy's integral formula: index of a closed path, nth derivative of a regular function
8) Series expansions: Taylor series, Laurent series, singularities of an analytic function
9) The residue theorem: calculation of residues, applications
10) Advanced topics: Riemann surfaces, analytic continuation, monodromy theorem, Schwarz's reflection principle
Theory Lectures and Exercises.
Classwork and Homework.
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.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Exercises | 14 | 0.56 | 1, 2, 3, 5, 6, 8 |
Theory Lectures | 27 | 1.08 | 1, 2, 3, 5, 6, 8 |
Type: Autonomous | |||
Discussion, Work Groups, Group Exercises | 19 | 0.76 | 1, 2, 4, 3, 5, 7, 9, 6, 8 |
Study of Theoretical Foundations | 36 | 1.44 | 1, 2, 4, 3, 5, 7, 9, 6, 8 |
Exam and delivery of exercises for topics 1, 2, 3, 4 and 5;
Exam and delivery of exercises for topics 6, 7, 8, 9 and 10;
Make-up exam: all topics;
In order to participate in the make-up exam you have to be evaluated of the two partial exams without requiring a minimal mark;
The make-up exam covers the whole subject;
You can come to the make-up exam to improve your mark. If so, your final mark corresponding to the exam part will be that of this exam.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Delivery of Exercises: topics 1, 2, 3, 4 and 5 | 10% | 10 | 0.4 | 4, 3, 5, 7, 9, 6, 8 |
Delivery of Exercises: topics 6, 7, 8, 9 and 10 | 10% | 10 | 0.4 | 1, 2, 4, 7, 9, 8 |
Exam: topics 1, 2, 3, 4 and 5 | 40% | 3 | 0.12 | 4, 3, 5, 7, 9, 6, 8 |
Exam: topics 6, 7, 8, 9 and 10 | 40% | 3 | 0.12 | 1, 2, 4, 7, 9, 8 |
Make-up Exam: all topics | 80% | 3 | 0.12 | 1, 2, 4, 3, 5, 7, 9, 6, 8 |
Bibliography: Complex Variables
• "Complex Variables", M. R. Spiegel et al., Schaum's Outline Series, McGraw-Hill
• "Complex Variable and Appications", J. W. Brown and R. V. Churchill, McGraw-Hill
Bibliografia: Fourier Series and Transform
• "Mathematical Methods for Physicists", G. B. Arfken and H. J. Weber, Elsevier Academic Press
It is recommended to use Mathematica Student Edition.