Dear Saddam Hussein Khuhro, 

Yes, when s = jw then the Laplace becomes the Fourier transform.

As you might already know, if we put the exp(-sigma t)  to multiply the Fourier transform, then it becomes the Laplace transform. We can easily see that what is written in the Fourier transform as exp(-jwt), now becomes exp(-st) where s = sigma + jw. The sigma is the real or called as the exponential part, where the jw is the imaginary or called sinusoidal part. The Fourier transform does not really care on the changing magnitudes of a signal, whereas the Laplace transform 'care' both the changing magnitudes (exponential) and the oscillation (sinusoidal) parts. We can say that Fourier transform is a subset of Laplace transform.

The Laplace transform is essentially helpful for solving differential equations since most of any differential equation's solution will contain exponential and sinusoidal parts. The solution can be more easily express and understand in the s domain.

Please have a look the following video.

https://www.youtube.com/watch?v=ZGPtPkTft8g

I hope this would help. 

Dear Sadam Hussain Khuhro ,

Both are actually different and used for different purpose. They couldn't be same at all.

1. Fourier transform is the special case of laplace transform which is evaluated keeping the real part zero.

2. Fourier transform is generally used for analysis in frequency domain whereas laplace transform is generally used for analysis in s-domain(it's not frequency domain).

3. Fourier transform helps us to study anything in the frequency domain whereas laplace transform is usually done for complex analysis (when anything is not easier to analyse in time domain, we convert it into s domain and then take the inverse laplace transform to complete the analysis).

PS: I have answered this question with respect to electronics engineering domain.

Dear Sadam Hussain Khuhro

They will not be the same if you put s=jw. But you have to change also the integration limit of the function f(t) from - infinity to + infinity instead of from 0 to +infinity.

For the physical meaning and the usefulness of the two domains please read the following:

The La Place transform is a tool to solve the linear differential equation in an easy and elegant way since it transforms a linear differential equation into algebraic equation.

In order to solve for the response of a system in a time domain upon an application of time domain stimuli, one must solve the linear system of the differential equations mathematically modelling the system together with the initial and boundary conditions. Such solution will be normally tedious.

But using the La Place transform to transform the equations from time domain into the S-domain, which is called the complex frequency domain, one can convert the time domain differential equations into S-domain algebraic equations.

Naturally simultaneously solving algebraic equations is much easier than solving simultaneous differential equations. This is the power of this mathematical transformation.

As for the Fourier transform, it is a tool for transforming time domain functions into frequency domain function. Physically any time domain function can be thought of a summation of complex exponential signals. And this what the Fourier transform yield. So, it is an analysis tool of time domain signals.

The Fourier transform is a subset of the La Place transform as the colleagues above pointed.

Best wishes