The Laplace transform usually takes 50 minutes of lecture in front of
a chalkboard to explain to students who already have 2 semesters of
calculus.
There won't be many takers on this for $3.
I'll say a little in the hopes that it helps a little.
The Laplace transform takes a continuous function and maps it in a
special way to a different function. Any continuous function that does
not grow faster then an exponential function has a Laplace transform
(whether or not you can compute it is another question.) . If f(t) was
our original function then let F(s) represent its laplace transform. A
thing that teachers say, is that you are bringing the function from
the "t domain" to the "s domain." The function won't look anything
like it used to, but there is a procedure for mapping them back and
forth. Ok, so the definition of the laplace transform of f(t) is
L{f(t)} = F(s) = integral ( f(t)*exp(-st) dt from 0 to infinity) Like
I said, so long as the f(t) doesnt grow faster then an exponential
function, this integral will converge.
SO, very simple example
Laplace Transform of f(t) = 1 (a constant function) is
integral ( 1*exp(-st) from 0 to infinity) = 1/s (I am presuming you
have 2 semesters equivalent of calculus.)
other simple ones can be computed,
L{t^n} = n!/(s^(n+1))
L{sin(bt)} = b / (s^2+b^2)
L{cos(bt)} = s / (s^2+b^2)
Laplace transforms are good for differential equations, because...
basically, you can transform the functions in a given differential
equation with the laplace transform, and certain families of really
hard to calculate things become much easier to calculate.. and then
you can transform them back.
The z-transform is very simmilar to the laplace transform, except it
transforms discrete functions (also called sequences) in a very
simmilar manner.
let f_n (written f subscript n) be a sequence,
then its z-tranform is sum f_n * z^n from 0, infinity.
So any sequence that grows no faster then geometric can be tranformed.
Who cares about z-transforms?, MOSTLY electrical engineering students
who are taking classes in signals and systems; also people studiyng
advanced probability although they would call it "a generating
function" of a discrete random variable. |