Second Shifting theorem to find Laplace transform

In summary, the conversation discusses using the second shifting theorem to obtain the Laplace transform for a given function. The theorem states that L(g(t-k)u(t-k)) = e-skG(s), where u(t-a) is the heaviside step function. The process involves adding and subtracting an expression to account for the function being equal to zero for t<a and equal to 1 for t>a. However, there is no "proper" way to do this according to the lecture notes.
  • #1
helpinghand
39
0
Can some one help me to understand this:

I need to use the second shifting theorem to get the Laplace transform, given:

f(t) = { 4 - t2 , t < 2 ...... 1
{ 0 , t >= 2


They got this:

f(t) = (4 - t2)(1 - u(t - 2)) .... 2


I know that the second shifting theroem says that L(g(t-k)u(t-k)) = e-skG(s)

But the thing is I don't know how they get from 1 to 2, can someone please explain this?

Cheers
 
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  • #2
Mark's notes leave a bit to be desired

you have to use the heaviside step function u(t-a)
I think of it as a function that can be switched on at any time t and remains on forever
knowing that the function equals zero for t<a and equals 1 for t>a.
We want the function to be equal to 4-t^2 for t less than 2. but u(t-2)=0 for t<2
so we must add the expression 4-t^2 for t<2, then take it away for t>2

hence:
f(t)=(4-t^2)-(4-t^2)u(t-2)

As far as i can see there is no "proper" way to do this, certainly not in the lecture notes anyway
 

1. What is the Second Shifting theorem?

The Second Shifting theorem is a rule that allows us to find the Laplace transform of a function that has been shifted horizontally in the time domain. It states that if we have a function f(t) with a Laplace transform F(s), and we shift the function by a time t0, the Laplace transform of the shifted function is e-st0F(s).

2. How do I use the Second Shifting theorem?

To use the Second Shifting theorem, you must first have a function in the time domain and its corresponding Laplace transform. Then, if the function is shifted by a time t0, the Laplace transform of the shifted function can be found by multiplying the original transform by e-st0. This allows us to easily find the Laplace transform of functions that have been shifted in time.

3. When is the Second Shifting theorem useful?

The Second Shifting theorem is useful in cases where we need to find the Laplace transform of a function that is shifted in time. This often occurs in real-world applications, such as in electrical engineering, where signals may be shifted due to delays or other factors. It allows us to easily find the Laplace transform of the shifted function without having to go through the entire process again.

4. Can the Second Shifting theorem be used for any type of function?

Yes, the Second Shifting theorem can be used for any type of function as long as it has a Laplace transform. This includes piecewise, periodic, and exponential functions. It is a general rule that can be applied to any function that has been shifted in time.

5. Are there any limitations to the Second Shifting theorem?

The Second Shifting theorem can only be used for horizontal shifts in the time domain. It cannot be applied to vertical shifts or other types of transformations. Additionally, it only applies to functions that have a Laplace transform. If a function does not have a Laplace transform, the Second Shifting theorem cannot be used to find its transform.

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