Checking date: 12/05/2022

Course: 2022/2023

Complex variable and transforms
Study: Dual Bachelor in Engineering Physics and Industrial Technologies Engineering (370)


Department assigned to the subject: Department of Mathematics

Type: Compulsory
ECTS Credits: 6.0 ECTS


Requirements (Subjects that are assumed to be known)
Calculus II Differential Equations
CB1. Students have demonstrated knowledge and understanding in a field of study that builds upon their general secondary education, and is typically at a level that, whilst supported by advanced textbooks, includes some aspects that will be informed by knowledge of the forefront of their field of study CB2. Students can apply their knowledge and understanding in a manner that indicates a professional approach to their work or vocation, and have competences typically demonstrated through devising and sustaining arguments and solving problems within their field of study CB3. Students have the ability to gather and interpret relevant data (usually within their field of study) to inform judgments that include reflection on relevant social, scientific or ethical issues CB4. Students can communicate information, ideas, problems and solutions to both specialist and non-specialist audiences CB5. Students have developed those learning skills that are necessary for them to continue to undertake further study with a high degree of autonomy CG2. Learn new methods and technologies from basic scientific and technical knowledge, and being able to adapt to new situations. CG3. Solve problems with initiative, decision making, creativity, and communicate and transmit knowledge, skills and abilities, understanding the ethical, social and professional responsibility of the engineering activity. Capacity for leadership, innovation and entrepreneurial spirit. CG4. Solve mathematical, physical, chemical, biological and technological problems that may arise within the framework of the applications of quantum technologies, nanotechnology, biology, micro- and nano-electronics and photonics in various fields of engineering. CG5. Use the theoretical and practical knowledge acquired in the definition, approach and resolution of problems in the framework of the exercise of their profession. CE1. Solve mathematical problems that may arise in engineering and apply knowledge of linear algebra, differential and integral calculus, numerical methods, numerical algorithms, statistics, differential equations and in partial derivatives, complex and transformed variables. CE4. Analyze and manipulate analog and digital signals in the temporal and frequency domains, and understand and master the basic concepts of linear systems and related functions and transforms, as well as apply them to circuit design. CE22. Design, plan and estimate the costs of an engineering project. CT1. Work in multidisciplinary and international teams as well as organize and plan work making the right decisions based on available information, gathering and interpreting relevant data to make judgments and critical thinking within the area of study. RA1. To have acquired sufficient knowledge and proved a sufficiently deep comprehension of the basic principles, both theoretical and practical, and methodology of the more important fields in science and technology as to be able to work successfully in them; RA2. To be able, using arguments, strategies and procedures developed by themselves, to apply their knowledge and abilities to the successful solution of complex technological problems that require creating and innovative thinking; RA3. To be able to search for, collect and interpret relevant information and data to back up their conclusions including, whenever needed, the consideration of any social, scientific and ethical aspects relevant in their field of study; RA6. To be aware of their own shortcomings and formative needs in their field of specialty, and to be able to plan and organize their own training with a high degree of independence
Skills and learning outcomes
Description of contents: programme
1. Complex functions Complex numbers. Complex functions. Limits. Continuous functions. Derivatives and Cauchy-Riemann equations. Armonic functions. 2. Elementary functions Polynomials. Exponential function. Trigonometric functions. Hyperbolic functions. Logarithm. Complex exponents. Inverses of trigonometric and hyperbolic functions. 3. Integrals in the complex domain. Contour integrales. Cauchy-Goursat theorem. Cauchy formula. Morera theorem. Bounds for analytic functions. The fundamental theorem of algebra. 4. Series Sequences and convergence criteria. Power series. radius of convergence. Taylor series. Laurent series. Analytic continuation. Power series and differential equations. Frobenius theory. Special functions of Mathematical Physics 5. Residues and poles Zeros of a function. Singularities. Poles. Residue formula. Residue theorem. Real integrals of trigonometric functions. Real improper integrals. Integrals on branch cuts. Summations of series by using residue theorem. 6. Fourier series Fourier series and their application to periodic signals. Square integrable functions. Pointwise convergence. Uniform convergence. Application to differential and partial differential equations. 7. Fourier transform. Definition and properties. Inverse Fourier transform. Representation of aperiodic signals. Discrete time Fourier transform. 8. Laplace transform Definition, properties and convergence. Inverse Laplace transform. Derivatives, integrals, and convolution. Applications to systems of linear differential equations. Transfer function. 9. z-Transform Convergence region and other properties. Inverse z-transform. Transforms between continuous and discrete time signals. Applications to linear difference equations. Linear time-invariant (LTI) systems. Analysis of LTI systems with transforms.
Learning activities and methodology
AF1. THEORETICAL-PRACTICAL CLASSES. Knowledge and concepts students mustacquire. Receive course notes and will have basic reference texts.Students partake in exercises to resolve practical problems. It entails 44 hours with an 100% on-site. AF2. TUTORING SESSIONS. Individualized attendance (individual tutoring) or in-group (group tutoring) for students with a teacher.Subjects with 6 credits have 4 hours of tutoring/ 100% on- site attendance. AF3. STUDENT INDIVIDUAL WORK OR GROUP WORK.Subjects with 6 credits have 98 hours/0% on-site. AF9. FINAL EXAM. Global assessment of knowledge, skills and capacities acquired throughout the course. It entails 4 hours/100% on-site MD1. THEORY CLASS. Classroom presentations by the teacher with IT and audiovisual support in which the subject`s main concepts are developed, while providing material and bibliography to complement student learning MD2. PRACTICAL CLASS. Resolution of practical cases and problem, posed by the teacher, and carried out individually or in a group MD3. TUTORING SESSIONS. Individualized attendance (individual tutoring sessions) or in-group (group tutoring sessions) for students with teacher as tutor. Subjects with 6 credits have 4 hours of tutoring/100% on-site. MD6. LABORATORY PRACTICAL SESSIONS. Applied/experimental learning/teaching in workshops and laboratories under the tutor's supervi
Assessment System
  • % end-of-term-examination 40
  • % of continuous assessment (assigments, laboratory, practicals...) 60
Calendar of Continuous assessment
Basic Bibliography
  • A. Papoulis. Signal Analysis. McGraw Hill International editions. 1984
  • B. Fornberg, C. Piret. Complex Variables and Analytic Functions: An Illustrated Introduction. SIAM. 2019
  • D. Pestana, J. M. Rodríguez, F. Marcellán. Curso Práctico de variable compleja y teoría de transformadas. Pearson. 2014
  • J. W. Brown, R. V. Churchill. Complex Variables and Applications. McGraw Hill. 2009
Additional Bibliography
  • A. V. Oppenheim, A. S. Willsky, I. T. Young. Signals and Systems. Prentice Hall International Editors. 1983
  • I. Volkovyski, G. Lunts, I. Aramanovich. Problemas sobre la teoría de funciones de variable compleja. Mir. 1972
  • J. Bruna, J. Cufi. Complex Analysis. EMS Textbooks in Mathematics, European Mathematical Society. 2013
  • J. G. Proakis, D. G. Manolakis. Introduction to Digital Signal Processing. McMillan Publishing Company. 1988
  • P. Henrici. Applied and Computational Complex Analysis (3 volumenes). Wiley Classical Library, Wiley Interscience. 1993

The course syllabus may change due academic events or other reasons.