The pulse, step and exponential function
----------------------------------------

The Laplace transform of the pulse function $\delta(t)$ is:

.. math::
   X(s) = \int_{0}^{\infty} \delta(t) e^{-st} dt = 1 

The Laplace transform of the step function $u(t)$ is:

.. math::
   X(s) = \int_{0}^{\infty} u(t) e^{-st} dt = \int_{0}^{\infty}
   e^{-st}dt = \left[ -\frac{1}{s} e^{-st}\right]_0^{\infty}
   =-\frac{1}{s} \left(0-1\right) = \frac{1}{s}

Consider the exponential function

.. math::
   x(t) = A e^{a t}

this function has Laplace transform:

.. math::
   X(s) = \frac{A}{s+a} 

Summarizing we have:

.. math::
   \delta(t) &\ltarrow 1 \\
   u(t) &\ltarrow \frac{1}{s} \\
   A e^{at} &\ltarrow \frac{A}{s+a}

Observe that the pulse is the derivative of the step function and the
Laplace transform of the pulse is $s$ times the Laplace transform of
the step function. This relation between the \Lt s of a function and
its derivative is valid in general.