Fourier transforms
In the previous section, we learned about Fourier series, and how they allow us to decompose, or equivalently build up, periodic functions in terms of simple building blocks – sine and cosine waves.
But what if our function is not periodic? If a function is not periodic, there is no finite value of 
over which the function repeats itself, so effectively, is infinite. What happens if we make go to infinity in our Fourier series in Eq. 25? The sums in Eq. 25 become integrals in the limit, 
, and our Fourier series becomes as follows:
Eq. 28
We are still representing our function, 
, as a superposition of sine and cosine waves, but there are a couple of comments we need to make about Eq. 28:
- Note how the integration variable,
, is playing the equivalent role to the summation index in Eq. 25. - We have reverted to including sine and cosine waves with values of
between 
and 
, while in Eq. 25, we absorbed the cosine wave amplitudes from 
...
