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Fourier Transforms

Using the -function we may now complete the integral in (10),

The integral appears as the product of the two functions in Figure 3.4.2.

 

Because the peak may be made arbitrarily narrow, the values of u which contribute to the integral are so tiny that we may make the approximation with a small error proportional to at most and which will vanish in the final limit . Thus, we have

Our initial guess was nearly correct! We have found

or

This tells us that if we take the superposition

we may find from from the inverse relationship

(12) and (13) taken together are the Fourier inversion theorem, where the operation in (12), is known as the Fourier transform, while the operation in (13), is the inverse Fourier transform.



Prof. Tomas Alberto Arias
Wed Oct 11 21:10:55 EDT 1995