Linear Algebra, Calculus I, Calculus II

A.1. To understand the concept of complex analyticity. A.2. To be able o compute the Laurent or Taylor series expansions associated to a function which is analytic in part of the complex plane, and to determine the region of convergence of such series. A.3. To acquire the basic concepts related to the elementary complex functions. A.4. To compute definite integrals by means of the residue calculus. A.5. To understand and solve first and second order linear homogeneous and non-homogeneous differential equations. A.6. To solve second order equations using power series methods. A.7. To recognisee classical PDEs describing physical processes such as diffusion, wave propagation and electrostatics. A.8. To solve analytically, using the method of separation of variables, the heat and wave equations (in one space variable). B.1. To understand the concept of complex differentiation and its practical applications. B.2. To be able to handle functions given in terms of series. B.3. To understand the concept of concept of complex integration and its practical applications. B.4. To be able to solve first and second order linear homogeneous and non-homogeneous ODEs. B.5. To be able to solve second order ODEs using power series methods. B.5. To be able to model real-world problems using PDEs, and solve them using Fourier techniques. C.1. To be able to think abstractly, and to use induction and deduction. C.2. To be able to communicate in oral and written forms using appropriately mathematical language. C.3. To be able to model a real situation using differential equation techniques. C.4. To be able to interpret a mathematical solution of a given problem, its accuracy, and its limitations.

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 CG3: Knowledge of basic and technological subject areas which enable acquisition of new methods and technologies, as well as endowing the technical engineer with the versatility necessary to adapt to any new situation. CG10: Ability to solve mathematical problems arising in engineering. Aptitude for applied knowledge in: linear algebra, geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives; numerical methods; numerical algorithms; statistics and optimization. RA1: Knowledge and understanding of the general fundamentals of engineering, scientific and mathematical principles, as well as those of their branch or specialty, including some knowledge at the forefront of their field.

1. COMPLEX ANALYSIS. Analytic functions and singularities. Laurent series. Contour integration and Cauchy's integral formula. The residue theorem and its applications. 2. ORDINARY DIFFERENTIAL EQUATIONS. First order equations. Second order linear equations. Power series solutions and special functions. Fourier series solutions of ODEs. The Laplace transform: Applications to differential equations. 3. PARTIAL DIFFERENTIAL EQUATIONS. Heat, wave, and Laplace equations. Fourier's method of separation of variables.

Lecture sessions: 3 credits. Problem sessions: 3 credits.