Laplace transform
In
mathematics and in particular, in
functional analysis, the
Laplace transform of a
function f(
t) defined for all
real numbers t ≥ 0 is the function
F(
s), defined by:

The transform has a number of properties that make it useful for analysing linear dynamic systems. The most significant advantage is that
integration and
differentiation become multiplication and division. (This is similar to the way that logarithms change multiplication of numbers to addition.) This changes integral equations and
differential equations to polynomial equations, which are much easier to solve. The inverse is the
Bromwich integral, which is a
complex integral.
Also, the output of a linear dynamic system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication, which often makes matters easier. For more information, see control theory.
The Laplace transform is named in honor of PierreSimon Laplace.
A sometimes convenient abuse of notation, prevailing especially among engineers and physicists, writes this in the following form:

When one talks about the Laplace transform, one is generally referring to the unilateral version. There also exists a bilateral Laplace transform, which is defined as follows:

The Laplace transform
F(
s) typically exists for all real numbers
s >
a, where
a is a constant which depends on the growth behavior of
f(
t).
The Laplace transform can also be used to solve differential equations.

Differentiation





Integration

s shifting


t shifting


Note: is the step function.

Laplace transform of a function with period p

See also