The Fourier Transform is merely a restatement of the Fourier Integral:

Using the complex form of Cosine, we can easily prove that the above integral can be re-written as:

The above integral can be expressed by the following Fourier Transform pair:

Since is a dummy variable, we can replace it with and define the Fourier transform of and its inverse transform as:

where and are the Fourier and its inverse transform operators, respectively.

If is an even function, then its Fourier Integral is equivalent to the following pair of equations:

where is called the Fourier Cosine Transform operator.

On the other hand, if is an odd function, then its Fourier Integral is equivalent to the following pair of equations:

where is called the Fourier Sine Transform operator.