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 (ODEs) 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. To that end, Laplace transformation is used to solve initial-value problems (IVPs)—ODEs subjected to initial conditions—by transforming the data from time domain to the s -domain, where equations are algebraic and hence easier to work with. Transformation of the information from the s -domain back to time domain ultimately describes the solution of the IVP.