What are the several definitions you are referring to?

Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes (and phase) of the individual notes that make it up.

The Fourier transform has many applications, in fact any field of physical science that uses sinusoidal signals, such as engineering, physics, applied mathematics, and chemistry, will make use of Fourier series and Fourier transforms. It would be impossible to give examples of all the areas where the Fourier transform is involved, but here are some examples from physics, engineering, and signal processing:

> Communications

> Astronomy

> Geology

> Optics

Attached is a link from wolfram that gives you a mathematical description.

http://mathworld.wolfram.com/FourierTransform.html

I mean the constant can be varied for Fourier Transform. For example, in one book, the constant is one. However, in another textbook, the constant is 1/the square root of 2pi. It doesn't like Laplace Transform definition which is always

same. That is the integral of exp^(-st) f(t) dt from 0 to infinity.

Fourier transforms to heat and mass transforms

Fourier transform is in fact a subset of Laplace transform where s is replaced by jw

There are, in principle, three different definitions of the (integral) Fourier transform known the literature. The only difference is their exp function exponent and the resulting factors in front of the integral: 1/(2 pi), 2 pi, sqrt(2 pi). Mostly, a factor of 2 pi is missing in the exp function exponent and, hence, factors of 1/(2 pi) and 2 pi appear in front of the integral in order to compensate that. For symmetry reasons, the version employing a factor of sqrt(2 pi) is widely used. However, the best definition is the one used at http://en.wikipedia.org/wiki/Fourier_transform. This one is truly symmetrical: time and spectral domain are fully equivalent and there are no disturbing factors while forward of backward transforming. The only difference to the two before mentioned definitions is that 1-periodic functions are used as function basis instead of 2-pi-periodic functions. In engineering, however, might be a physical requirement to choose 2 pi - periodic functions as function basis.

The easiest way of choosing one is to substitute 2.pi.f for omega. If you use the 2.pi and put it in the inverse, it will disappear when making the substitution. The formulae (direct and inverse) become almost equal. On the other hand and as referred by other colleague, this FT will be a particular case of the two-sided Laplace Transform.

Thank you. It really gives me enlightment.

Thank you, Henk Smid.

Read any book on Signals and Systems.

In addition to my previous answer, let me mention that "There is only one Fourier Transform". I have shown this recently in https://doi.org/10.13140/RG.2.2.30950.83521. If you use the "Fourier transform in the tempered distributions sense" it'll be fine for any application. 

I agree with Jens, but I call the attention to the fact that we do not need to start from a continuous formulation. The discrete-time signals and systems exist and are the base for many of our daily applications.

The Ugur's definition is so good that is useless.

Thank you Professor Ortigueira. I agree, all four Fourier transform definitions have their justifications. It is, however, good to know that they coincide finally in some sense.