# Analysis of linear systems in extended frequency space

The description of linear system will be extended into the complex frequency domain in the following chapters, in order to investigate what kind of association may be created between the various system responses.

Definition: Let real variable and real as well as complex value general step function:

, as well as

operation is the Laplace transformation of . Sufficient condition of Laplace transformation is the function should be absolutely integrable (the condition is not necessary, but sufficient).

Laplace transformation is linear, i.e.:

, where

Let’s see, the relation between the Laplace and Fourier transformation. It is known from the above mentioned, the sufficient condition of both Laplace and Fourier transformation is , i.e. should be absolutely integrable.

Fourier transformation:

Laplace transformation:

Apply the following substitution in the Laplace transformation expression: .

It is known from the definition of Laplace transformation, that should be general step function. If is general step function, then is general step function too.

Consequently

The conclusions are the followings:
Under the conditions, that is general step function and absolutely integrable, the Laplace transformation of among complex axis (i.e. among the imaginary axis) gives the Fourier transformation. Consequently, it is possible to see, the domain of Fourier transformation is the imaginary axis, while the domain of Laplace transformation is the complete complex plane. Another approach is existing also, that the Laplace transformation of is possible to derive from the Fourier transformation of function. Consequently can be expressed by the inverse Fourier transformation as follow:

where the expression of inverse Laplace transformation is obtained,
and called Rieman-Melling integral.