Definition

The Laplace transform is an important tool for solving systems of linear differential equations with constant coefficients. The differential equations to be solved for control tasks normally fulfil the conditions that must be met for taking the Laplace transform. The Laplace transform is an integral transformation, which maps a large class of original functions in the time domain unambiguously reversible into image functions in the domain. This mapping is performed via the Laplace integral of , that is

where in the argument of the Laplace transform the complex variable appears. For the application of Eq. (2.1) to causal systems considered here the following two conditions for the time function must be met:

1. for ;
2. the integral in Eq. (2.1) must converge.

To show the correspondence between the original and mapped functions it is useful to use the operator notation

Another possibility of correspondence is to use the sign in the following way:

During the treatment of control systems usually the original function is a function of time. As the complex variable contains the frequency , the image function will often be called a frequency function. Therefore, the Laplace transform allows one to make a transition from the 'time domain' into the 'frequency domain' according to Eq. (2.1).