Complex analysis and transforms

Instruction is in the form of lectures and exercises. The course includes the following components: Series: <br> Series of real numbers. Series of functions. Fourier series. Different types of convergence. Convergence criteria. Complex analysis: <br> The field of complex numbers. Elementary functions: Complex exponential function, complex logarithmic function, complex trigonometric and hyperbolic functions. Real and complex differentiability, Cauchy-Riemann equations, analyticity of complex function Ln, power functions. Complex integration, the ML inequality and its consequences, Cauchy integration formula. Leibniz-Newton theorem. Power series. Abel theorem. Cauchy-Hadamard theorem in complex analysis. Analytical functions in circular regions. Laurent series and residue. Isolated singular points of analytical functions and the residue theorem. Calculation of certain real improper integrals with the residue theorem. Cauchy principal value for improper integrals and their calculation with the residue theorem. Transformation theory: <br> Laplace transformation and its basic applications in solving differential equations and systems of differential equations with constant coefficients. Determination of inverse Laplace transformations with the residue theorem. Fourier transformation and some applications of this transformation in certain types of partial differential equations.