The Laplace transform maps a function of to a function of We define
, where is a constant.
(This is proved later in the wiki.)
Note: Here, we are transforming a function of into a function of
|where is the gamma function.|
What we have in the integral is instead of , so the function gets shifted.
Differentiate with respect to times to get
Some examples are shown here, which demonstrate how to calculate the Laplace transform of some given functions.
First, split it as two Laplace transforms:
Now, we know and Since the exponential function shifts the Laplace transform,
which is the answer.
A neat trick to find the is puttin ; then the becomes zero and the integral also becomes zero, so we have
If then the inverse Laplace transform of is .
We see some examples of how to calculate Laplace inverse.
We can ignore the constant (in this case 5) as it doesn't affect our Laplace transform much. We can factor the denominator into easy linear factors, so let's try that
We use the fact (proved earlier), so this becomes
We first note that
Since we are shifting the LHS by 9, we multiply by to get
We can write this as
By doing what we did in the last problem, we have this to be equal to
The convolution theorem for Laplace transform is a useful tool for solving certain Laplace transforms. First, we must define convolution.
The convolution of two functions is given by
Here is an example of convolution:
Find the convolution
We use the definition
We can use trigonometric identities to write this as
Both integrals can be solved using double- or half-angle identities to get
Plugging in double-angle identities, we will get the last two terms to cancel out, which gives us
This will be useful in Laplace transforms because of the convolution theorem:
The convolution theorem states that
Here make a substitution:
Then the integral turns into
We first start with the following theorem:
We see the base case at is true by using integration by parts. So assume this is true for then
Now consider , then
by induction. Hence proved.
How do we use this? FIrst note that in a simple form and as our examples are mainly ODEs.
Solve the ODE
First take the Laplace transform of both sides:
Taking Laplace inverse,
Consider the right integral:
Changing the order of integration and performing inner integration on variable we get the result
So let's see how to apply this:
Prove the Dirichlet integral
This famous integral can be proved in one line:
Even without this, we can solve some integrals like:
So, a seemingly difficult integral that would have taken forever with tabular integration is solved in less than 5 minutes with Laplace transform.