# Thread: Laplace transform and Fourier transform what is the different?

1. ## Laplace transform and Fourier transform what is the different?

Hello,
I want to know what is the different between Laplace transform and Fourier transform?
Can anyone show me? and help me about it?

2. Well, for one, they don't have the same definition. What else do you want to know? You have to be specific.

3. Originally Posted by Hamed
Hello,
I want to know what is the different between Laplace transform and Fourier transform?
Can anyone show me? and help me about it?
They have a different kernel. In Laplace, the kernel is $e^{-st}$. In Fourier, the kernel is $e^{-2\pi i x t}$

Also, they are defined over different domains. Laplace is calculated for $t>0$, where as in Fourier, $t\in(-\infty,\infty)$.

As Bruno said, please be more specific in what you're asking. These are just a couple of the differences between these two transforms.

4. My master told me they are same frequency domain and I said no they are different and Fourier is frequency domain!
What is Laplace domain?

5. Originally Posted by Hamed
My master told me they are same frequency domain and I said no they are different and Fourier is frequency domain!
What is Laplace domain?
For an absolutely integrable real function $f(x)$ which is zero for all $x<0$, then $(Ff)(\omega)=(Lf)(i \omega)$ (assuming I have done the algebra right and give or take a multiplicative constant due to how the FT is defined)

CB

But I need show diffrent
In wiki I read this :
The Laplace transform is related to the Fourier transform, but whereas the Fourier transform resolves a function or signal into its modes of vibration, the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations
Can anyone explain it more for me?

7. Originally Posted by Hamed
But I need show diffrent
In wiki I read this :

Can anyone explain it more for me?
Exactly what do you not understand in the phrase "the Fourier transform resolves a function or signal into its modes of vibration"?

And the phrase "the Laplace transform resolves a function into its moments" (you can Google for the definition of what the moments of a function are if need be)

Both are of use in differential equations because of the relationship between the FT and LT of a derivative and the function itself, and certain other properties which the two transforms have in common such as the convolution theorems, ...

CB

8. vibration and moments I can not find exact defenation!

9. Originally Posted by Hamed
vibration and moments I can not find exact defenation!
The Fourier transform gives a representation of a function in terms of a linear supposition of sinusoids, that it analyses the function/signal into vibrational components.

The Laplace transform is related to the moment generating function, and allows the generation of the moments when they exist (in fact they both do).

CB