Question

what is the difference between laplace and fourier transform

Answer 1

Here is a heuristic point of view from engineering considerations. I must confess I do not fully know the mathematical reasons.

Suppose you want to consider f(t), a function of time, t. Imagine that as we look at the direction of positive t-axis, the graph of f(t) s like looking behind to the trail f left in time. If you do not care about the future, ie the case t<0, then it makes sense to use Laplace transform, because the transform integral goes from 0 to ∞. On the other hand, if you care about the future also, it makes more sense to consider the Fourier transform. The transformation integral here goes from −∞ to ∞.

So if you want to include future in your analysis, then Fourier transform is the way. This makes sense in electrical engineering applications for example, where you consider sinusoidal signals and you have an idea of what is going to come.

However for some physical systems, you only have the data of what happened until then. And you want all your analysis to be based on this, without predicting the future. Then Laplace transforms is the way.

If you do not care about the future, ie if you can declare f(t)=0 for t<0, then the Laplace and Fourier transforms coincide: The Fourier transform is nothing but the Laplace transform evaluated on the imaginary axis. Such systems are called causal systems: the response depends only on what happened so far. This a terminology from control systems or signal processing.

For control systems engineering, stability of electrical networks, etc., Laplace transformation defines a more natural transfer function, and is easier to deal with, and the poles and zeros would immediately tell you about the stability of the network under consideration. Here we use Laplace transforms rather than Fourier, since its integral is simpler.

For instances where you look at the "frequency components", "spectrum", etc., Fourier analysis is always the best. The Fourier transform is simply the frequency spectrum of a signal. If you know that the sin/cos/complex exponentials would behave nicely, you might as well want to express a function in terms of these and observe how it behaves then.

Answer 2

The difference between the two is their domain
Laplace - time domain
Fourier - frequency domain
Generally Laplace is used in circuits, filters, and control system analysis. Fourier are used in communications, filters, and mineral explorations use instruments use FFT, Fast Fourier Transforms. coming under the category of spectral analysis.

Answer 3

It's simple,
The Laplace Transform is in s-domain where, \[s = \sigma+j \omega\] While, Fourier Transform is also in s-domain with \[\sigma = 0\] so is called in j-omega domain or frequency domain. Here, sigma is known as neper frequency and represents the exponential characteristic of the signal i.e. exponentially rising or falling.
While Fourier Transform only gives the frequency response of a system or frequency domain representation of a signal; Laplace Transform gives the complete system response (Transfer Function) or signal representation in complex s-plane with it's frequency component as well as time domain characteristic of exponential fall or rise or steady state (at sigma = 0).
In language of typical mathematics, While Fourier Transform expresses a function or signal in series of modes of vibrations (frequencies), Laplace Transform resolves the function into it's moment. Moment can be defined as a quantitative measure of shape of a set of points in simple terms.
I guess it is helpful explanation.