Properties of the Laplace Transform
DOI link for Properties of the Laplace Transform
Properties of the Laplace Transform book
This chapter concerns the Laplace transform (LT), which provides an extremely powerful tool for analyzing linear systems. Recall that the power and usefulness of the phasor approach is in transforming linear, ordinary differential equations with constant coefficients to algebraic equations for analyzing the sinusoidal steady state only. The LT extends this approach by transforming linear, ordinary differential equations also to algebraic equations, but for deriving the complete response, that is, steady-state plus transient. Moreover, the excitation is not limited to sinusoidal excitation but could be any arbitrary excitation that has an LT. The chapter introduces the LT before presenting its most important properties. This is followed by a discussion of the solution of linear, ordinary differential equations using the LT, and the inversion of the LT to obtain the time function. Of special interest are the initial-value theorem and the convolution theorem. The latter theorem provides the important link between the convolution integral and the LT.