Laplace Transformations:
Laplace Transformation (LT), Inverse Laplace Transforms (ILT), Applications of LT to de, Complex Variable Theory: complex number system, polar form of complex numbers, operations in polar form, De Moivre’s theorem, roots of complex numbers, functions, limit, continuity, derivatives, Cauchy-Riemann equations, line integral formulas, Green’s theorem in the plane, Integrals, Cauchy’s theorem, Cauchy’s integral formulas, Taylor’s series, Singular points, Poles, Laurent’s series, residues, residues theorem, evaluation of definite integrals, Convergency and Divergency of infinite series: general properties of series, test of convergence and divergence of the series of constants, power series, uniform convergence, Cauchy criterion for uniform convergence. Power series solution of de: analytic and singular functions, ordinary and singular points of a differential equation, Solution in series of the de.
Fourier Analysis:
Beta and Gamma functions, Fourier series and Applications: Periodic functions and Trigonometric series, Definition of Fourier Series, Dirichlet conditions, Fourier sine and cosine series, Half-Range Fourier Sine and Cosine series, Perseval’s Identity, Integration and differentiation of Fourier series, Complex notation for fourier series, Some physical applications of fourier series, Fourier Integrals and Applications: Fourier integral, Fourier transforms, Fourier sine and cosine transforms, Persival’s Identities for Fourier Integrals, Convolution theorem for Fourier transforms, Applications of Fourier Integrals and transforms, Special functions: Legendere’s de and Legendere’s polynomials, Bessel’s de and Bessel functions, Hermite’s de and Hermit’s polynomials, Laguerre’s DE and Laguerre’s polynomials.