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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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COPYRIGHT © 1996 Cuthbert Nyack.