Complex Variable and Transforms Engineering & Social Sciences Program Fall 2021 Semester - Madrid

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.
10. Linear invariant-time systems
Linear time-invariant (LTI) systems. Analysis of LTI systems with transforms.