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Definition 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.

Fourier Cosine and Sine Transforms 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.

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