For an -periodic function , we write:
where .
Note that we are summing over .
Call .
Then .
This is known as the complex Fourier series of .
Parseval’s Identity for -periodic functions and is:
By writing and ,
we find the Fourier series of :
where and .
Call these , , then:
Convergence of Fourier series
Define:
Then if is continuous on
apart from finitely many jump discontinuities
and has a finite number of minima and maxima on
we have for all :
Sine and cosine series
For a function
we can define it’s even and odd extensions and .
These are then periodic functions
We can find their Fourier series in terms of cosines and sines respectively.
Hence we found a sine series for and a cosine series for .