Why do we integrate the Fourier transform? How do I use that area?

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The Fourier transform, used to transform signals between the domain of time (or space) and the domain of frequency, has many applications in physics and engineering.

It is an application that matches a function f with another function g defined as follows:

Where f is L¹, i.e. f has to be an integrable function in the sense of the Lebesgue integral. The factor, which accompanies the integral in definition facilitates the enunciation of some of the theorems referring to the Fourier transform. Although this way of normalizing the Fourier transform is the most commonly adopted, it is not universal. In practice, the variables x and ξ are usually associated with dimensions such as time -seconds- and frequency -hertz- respectively, if the alternative formula is used:

the constant β cancels the dimensions associated with the variables obtaining a dimensionless exponent.

The Fourier transform thus defined has a number of continuity properties that ensure that it can be extended to larger function spaces and even to generalized function spaces.