The Laplace Transform
The definition of the Laplace Transform that we will use is called a
"one-sided" (or unilateral) Laplace Transform and is given by:
The Laplace Transform seems, at first, to be a fairly abstract and esoteric
concept. In practice, it allows one to (more) easily solve a huge variety
of problems that involve linear systems, particularly differential equations. It allows for compact
representation of systems (via the "Transfer Function"), it simplifies
evaluation of the convolution integral, and it turns problems involving
differential equations into algebraic problems. As indicated by the quotes
in the animation above (from some students at Swarthmore College), it almost magically simplifies problems that otherwise
are very difficult to solve.
There are a few things to note about the Laplace Transform.
- The function f(t), which is a function of time, is transformed
to a function F(s). The function F(s) is a function of the Laplace
variable, "s." We call this a Laplace domain function.
So the Laplace Transform takes a time domain function, f(t), and converts it
into a Laplace domain function, F(s).
- We use a lowercase letter for the function in the time domain, and un
uppercase letter in the Laplace domain.
- We say that F(s) is the Laplace Transform of f(t),
or that f(t) is the inverse Laplace
Transform of F(s),
or that f(t) and F(s) are a Laplace
- For our purposes the time variable, t, and time domain functions will
always be real-valued. The Laplace variable, s, and Laplace domain
functions are complex.
- Since the integral goes from 0
to ∞, the time variable, t, must not occur in the Laplace
domain result (if it does, you made a mistake). Note that none of the
Laplace Transforms in the
table have the time
variable, t, in them.
- The lower limit on the integral is written as 0-. This indicates
that the lower limit of the integral is from just before t=0 (t=0-
indicates an infinitesimally small time before zero). This is a fine point,
but you will see that it is very important in two respects:
- It lets us deal with the impulse function, δ(t). If you
don't know anything about the impulse function yet, don't worry, we'll
discuss it in some detail later.
- It lets us consider the initial conditions of a system at t=0-.
These are often much simpler to find than the initial conditions at t=0+
(which are needed by some other techniques used to solve differential
- Since the lower limit is zero, we will only be interested in the
behavior of functions (and systems) for t≥0.
- You will sometimes see discussed the "two-sided" (or
bilateral) transform (with the lower limit
written as -∞) or a one-sided transform with the lower limit written as
0+. We will not use these forms and will not discuss them further.
- Since the upper limit of the integral is ∞, we must ask ourselves
if the Laplace Transform, F(s), even exists. It turns out that the
transform exists as long
as f(t) doesn't grow faster than an exponential function. This includes all functions of interest to us, so we will not
concern ourselves with existence.
Before we show how the Laplace Transform is useful, we need to lay some
groundwork. We start by finding the Laplace Transform of some functions
and from there move on to finding properties of the Laplace Transform. With tables of the Laplace Transform of
Functions and Properties
of the Laplace Transform it becomes possible to find the Laplace Transform
of almost any function of interest without resorting to the integral shown above. Applications of the Laplace Transform
are discussed next - mostly the use of the Laplace Transform to solve
differential equations. Finally, the inverse Laplace Transform is covered
(though this is a large enough topic that it has its