The S-Domain
============

The Laplace transform takes a continuous time signal and transforms it
to the $s$-domain.

The Laplace transform is a generalization of the CT Fourier
Transform. Let $X(s)$ be the Laplace transform of $x(t)$, then the
Fourier transform of $x$ is found as $X(j\omega)$.

For most engineers (and many fysicists) the Laplace transform is just
a mathematical trick to easily solve a class of partial differential
equations. The Laplace transforms a constant coefficient linear
differential equation relating the input $x(t)$ with the output $y(t)$
into an algebraic polynomial in $s$.  We will look at this class of
*constant coefficient linear partial differential equations*.

The dynamical behaviour of many (physical) systems are described with
a PDE in this class (like the inverted pendulum). This is no class
where you learn to model systems with PDE's. We just give some
examples.

In a later chapter on Control Theory the Laplace transform and its use
in solving these type of differential equations will be used quite a
lot.



.. toctree::

   laplace_definition
   laplace_fourier
   laplace_pulse_step_exp
   laplace_derivatives_integrals
   laplace_convolutions
   laplace_properties
   laplace_table
   laplace_differentialequations
   laplace_exercises