Laplace transforms: definition, properties, inverse Laplace transform, solving IVP's, Laplace transforms and special functions, convolution, impulses and the Dirac Delta functions. Special functions: Gamma, Beta, Error, trigonometric series, periodic extension and convergence, Fourier series, convergence of Fourier series, Integration and differentiation of Fourier series, conditions for the uniform of a Fourier series. Fourier transform, integral Fourier transform. Fourier sine and cosine transforms, applications to Fourier transform