Linear Algebra (Course : 1 Semester : 1), Differential Calculus (Course : 1 Semester : 1), Integral Calculus (Course : 1 Semester : 2), Vector Calculus (Course : 1 Semester : 2).

K03: Know the fundamental results of real, complex and functional mathematical analysis and how to apply them in solving theoretical and applied problems. S03: Apply mathematical language and abstract-rigorous reasoning in the enunciation and demonstration of results in various areas of mathematics. C07: Establish the definition of a new mathematical object, in terms of others already known for solving problems in different contexts.

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 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.