Mathematical Analysis III D
Code
7544
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Filipe Serra de Oliveira
Weekly hours
6
Teaching language
Português
Subject matter
Complex Analysis
1. Complex Numbers
General facts;
2. Introduction to Complex Functions
Complex functions representing isometries, dilations and reductions of the plane;
3. Complex Functions
Exponential function; logarithms and complex powers; circular and hyperbolic trigonometry;
4. Holomorphic functions
Differentiability in C; the Cauchy-Riemann conditions;
5. Analytic functions
Taylor expansions, power series and convergence radius. Link between analytic functions and holomorphic functions.
Ordinary Differential Equations
1. First order differential equations
Linear, separable and exact equations;
2. Second order differential equations;
Homogeneous and linear equations;
3. Systems of Linear Differential Equations with constant coefficients
4. Laplace Transform
Partial Differential Equations
1. Fourier Series;
2. Separation of variables method
Applications to the Wave equation and the heat equation.
Bibliography
Complex Analysis
J. Marsden and M.Hoffman, Basic Complex Analysis (Freeman, 1999).
R. Churchill, J. Brown and R. Verhey, Complex Variables and Applications (McGraw-Hill, 1976).
D. Zill and P. Shanahan, Complex Analysis with applications (Jones and Bartlett Publishers, 2003).
Differential Equations
J. Robinson, An introduction to ordinary differential equations, Cambridge university press.
Mattuck, A., Differential equations, Mit Open courseware
Teaching method
Theoretical classes consist on a theoretical exposition illustrated by examples of applications.
Practical classes consist on the solving of exercises of application of the methods and results presented in the theoretical classes.
Any questions or doubts will be adressed during the classes, during the weeekly sessions specially programmed to it or even at special sessions previously arranged between professors and students.