# Laplace transform and Fourier transform what is the different?

• December 28th 2010, 06:51 PM
Hamed
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?
• December 28th 2010, 06:58 PM
Bruno J.
Well, for one, they don't have the same definition. What else do you want to know? You have to be specific.
• December 28th 2010, 07:28 PM
Chris L T521
Quote:

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.
• December 28th 2010, 11:30 PM
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?
• December 29th 2010, 01:35 AM
CaptainBlack
Quote:

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
• December 29th 2010, 03:29 AM
Hamed
But I need show diffrent
In wiki I read this :
Quote:

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?
• December 29th 2010, 09:53 AM
CaptainBlack
Quote:

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
• December 29th 2010, 07:55 PM
Hamed
vibration and moments I can not find exact defenation!
• December 29th 2010, 10:51 PM
CaptainBlack
Quote:

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