Complex Analysis, Differential Equations, and Laplace Transformation

This chapter presents a review of complex analysis, differential equations, and Laplace transformation, providing the necessary background for a better understanding of various ideas and implementation of methods involved in the analysis of dynamic systems. Complex analysis comprises the study of complex numbers, complex variables, and complex functions. Ordinary differential equations arise in situations where the rate of change of a function with respect to its independent variable is involved. Differential equations are generally very difficult to solve, even for the simplest case of constant coefficients. Complex analysis consists of the study of complex numbers, complex variables, and complex functions and plays an important role in a wide range of areas in the analysis of dynamic systems, from the calculation of a system’s natural frequencies to the analysis of a system’s frequency response. Complex numbers have a two-dimensional character, because they consist of a real part and an imaginary part.