Fourier Transform, Cosine and Sine Transforms

Cuthbert Nyack

If the function f(t) one seeks to find the transform of is even then the exponent in the expression for the integral can be replaced with a cos. Also since the function is even then the integral for -ve t is the same as the integral for +ve t and one only needs to integrate for +ve t and multiply the result by 2. With this change the Fourier transform becomes the Cosine transform and is shown in the following 2 formulas.

When the angular frequency w variable is replaced by the cyclic frequency f, the the Cosine transform is represented by the following 2 formulas.

Sometimes the following definition with the same factors in front is used.

If the function f(t) is odd then a similar argument to the above produces the Sine transform.

Expressed in terms of the variable f this becomes.

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