Checking date: 22/04/2024

Course: 2024/2025

Complex Analysis
Bachelor in Applied Mathematics and Computing (Plan: 433 - Estudio: 362)

Coordinating teacher: CUESTA RUIZ, JOSE ANTONIO

Department assigned to the subject: Mathematics Department

Type: Compulsory
ECTS Credits: 6.0 ECTS


Requirements (Subjects that are assumed to be known)
Linear Algebra (Course : 1 Semester : 1), Differential Calculus (Course : 1 Semester : 1), Integral Calculus (Course : 1 Semester : 2), Vector Calculus (Course : 1 Semester : 2).
Skills and learning outcomes
CB1. Students have demonstrated possession and understanding of knowledge in an area of study that builds on the foundation of general secondary education, and is usually at a level that, while relying on advanced textbooks, also includes some aspects that involve knowledge from the cutting edge of their field of study. CB2. Students are able to apply their knowledge to their work or vocation in a professional manner and possess the competences usually demonstrated through the development and defence of arguments and problem solving within their field of study. CB3. Students have the ability to gather and interpret relevant data (usually within their field of study) in order to make judgements which include reflection on relevant social, scientific or ethical issues. CB4. Students should be able to communicate information, ideas, problems and solutions to both specialist and non-specialist audiences. CB5. Students will have developed the learning skills necessary to undertake further study with a high degree of autonomy. CG1. Students are able to demonstrate knowledge and understanding of concepts in mathematics, statistics and computation and to apply them to solve problems in science and engineering with an ability for analysis and synthesis. CG2. Students are able to formulate in mathematical language problems that arise in science, engineering, economy and other social sciences. CG4. Students are able to show that they can analyze and interpret, with help of computer science, the solutions obtained from problems associated to real world mathematical models, discriminating the most relevant behaviours for each application. CG5. Students can synthesize conclusions obtained from analysis of mathematical models coming from real world applications and they can communicate in verbal and written form in English language, in an clear and convincing way and with a language that is accessible to the general public. CG6. Students can search and use bibliographic resources, in physical or digital support, as they are needed to state and solve mathematically and computationally applied problems arising in new or unknown environments or with insufficient information. CE1. Students have shown that they know and understand the mathematical language and abstract-rigorous reasoning as well as to apply them to state and prove precise results in several areas in mathematics. CE2. Students have shown that they understand the fundamental results from real, complex and functional mathematical analysis. RA1. Students must have acquired advanced cutting-edge knowledge and demonstrated indepth understanding of the theoretical and practical aspects of working methodology in the area of applied mathematics and computing. RA2. Through sustained and well prepared argument and procedures, students will be able to apply their knowledge, their understanding and the capabilities to resolve problems in complex specialized professional and work areas requiring the use of creative and innovative ideas. RA3. Students must have the capacity to gather and interpret data and information on which they base their conclusions, including where relevant and necessary, reflections on matters of a social, scientific, and ethical nature in their field of study. RA5. Students must know how to communication with all types of audiences (specialized or not) their knowledge, methodology, ideas, problems and solutions in the area of their field of study in a clear and precise way. RA6. Students must be capable of identifying their own education and training needs in their field of study and the work or professional environment and organize their own learning with a high degree of autonomy in all types of contexts (structured or not).
Description of contents: programme
1. Introduction to complex numbers 1.1 Basic definitions 1.2 Polar representation 1.3 Powers and roots 1.4 Sets of C and special points 2. Holomorphic functions 2.1 Functions of a complex variable 2.2 Limits and continuity 2.3 Derivability and holomorphic functions: Cauchy-Riemann equations 2.4 Harmonic functions 3. Elementary functions 3.1 Exponential 3.2 Trigonometric functions 3.3 Hyperbolic functions 3.4 Logarithm 3.5 Complex powers 3.6 Inverse trigonometric and hyperbolic functions 4. Integration in the complex plane 4.1 Complex functions of a real variable 4.2 Contours 4.3 Contour integrals 4.4 Independence on contour: primitives 4.5 Cauchy-Goursat¿s theorem 4.6 Cauchy¿s integral formula 4.7 Applications of Cauchy¿s integral formula 5. Power series and analytic functions 5.1 Sequences and series of complex numbers 5.2 Sequences and series of complex functions 5.3 Power series 5.4 Analytic functions 5.5 Analytic continuation 5.6 Zeros and isolated singularities 5.7 Laurent¿s series 6. Integration by residues 6.1 Theorem of residues 6.2 Residues at poles 6.3 Residues at infinity 6.4 Calculation of real integrals through residues 6.5 Summing series with residues
Learning activities and methodology
LEARNING ACTIVITIES AND METHDOLOGY THEORETICAL-PRACTICAL CLASSES [44 hours with 100% classroom instruction, 1.76 ECTS] Knowledge and concepts students must acquire. Student receive course notes and will have basic reference texts to facilitate following the classes and carrying out follow up work. Students partake in exercises to resolve practical problems and participate in workshops and evaluation tests, all geared towards acquiring the necessary capabilities. TUTORING SESSIONS [4 hours of tutoring with 100% on-site attendance, 0.16 ECTS] Individualized attendance (individual tutoring) or in-group (group tutoring) for students with a teacher. STUDENT INDIVIDUAL WORK OR GROUP WORK [98 hours with 0 % on-site, 3.92 ECTS] FINAL EXAM [4 hours with 100% on site, 0.16 ECTS] Global assessment of knowledge, skills and capacities acquired throughout the course. METHODOS 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. PRACTICAL CLASS Resolution of practical cases and problems, posed by the teacher, and carried out individually or in a group. TUTORING SESSIONS Individualized attendance (individual tutoring sessions) or in-group (group tutoring sessions) for students with a teacher as tutor.
Assessment System
  • % end-of-term-examination 60
  • % of continuous assessment (assigments, laboratory, practicals...) 40

Calendar of Continuous assessment

Extraordinary call: regulations
Basic Bibliography
  • CHURCHILL, R.V. and BROWN, J.W.. Complex variables and applications. McGraw Hill. 1992
  • CHURCHILL, R.V. and BROWN, J.W.. Complex variables and applications: Selected Solutions to Exercises. McGraw Hill. 1992
  • LARS V. AHLFORS. Complex Analysis. McGraw Hill. 1979
  • LEVINSON, N. and REDHEFFER, R. M.. Complex Variables. Holden-Day. 1970
  • SPIEGEL, M.R.. Schaum's Outlines: Complex Variables. McGraw Hill. 1964
Additional Bibliography
  • PESTANA, D., RODRÍGUEZ, J.M. and MARCELLÁN, F.. Curso práctico de variable compleja y teoría de transformadas. Pearson. 2014

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