We can also show the the Fourier Series is a special case of the Fourier transform. We first note that (with the limits applied)
From these relationships it follows that 
where
and
In the limit at 
, we obtain the transform pair,
where 
and
That 
is obvious. To see that
note that
is a periodic function (in 
) with period
. As 
, in the interval
, 
tends toward 
.
Now consider a periodic function 
with period T. Define 
as,
Then,
Taking the transform both sides of this expression using the convolution
theorem and the transform pair 
,
Since 
we can write,
Thus, the Fourier transform consists of a weighted sum of 
functions at equally-spaced intervals. Taking the inverse transform of
this expression (noting that 
) we obtain the familiar Fourier series,
where
An interesting result of this derivation is that when computing 
it
may be simplier to compute 
and sample it at 
rather than directly computing 
from it's defintion.