In the study of Fourier Series, you will find that every continuous function f on an interval [-L,L] can be expressed on that interval as an infinite series of sines and cosines. For example, if the interval is [-p,p ],

The theory of Fourier series relies on the fact that the functions

Here, we will verify this fact.

We will use the following trigonometric identities:

We have six general integrals to evaluate to prove the orthogonality of the set {1, cosx, sinx, ¼}. In each of the following, we assume m and n are distinct positive integers.

1.  

   ó õ p -p  1·cos(nx) dx

   =

   1 n

   sin(nx) ô ô p -p = 0.

   
   
2.  

   ó õ p -p  1·sin(nx) dx

   =

   -

   1 n

   cos(nx) ô ô p -p = 0.

   
   
3.  

   ó õ p -p  sin(nx)cos(nx) dx

   =

   sin2(nx) 2n ô ô p -p = 0.

   
   
4.  

   ó õ p -p  sin(mx)sin(nx) dx	=	ó õ p -p  1 2 [cos(m-n)x-cos(m+n)x] dx
   	=	sin[(m-n)x] 2(m-n) - sin[(m+n)x] 2(m+n) ô ô ô p -p
   	=	0

   
   
5.  

   ó õ p p  cos(mx)cos(nx) dx	=	ó õ p -p  1 2 [cos(m-n)x+cos(m+n)x] dx
   	=	sin[(m-n)x] 2(m-n) - sin[(m+n)x] 2(m+n) ô ô ô p -p
   	=	0.

   
   
6.  

   ó õ p -p  sin(mx)cos(nx) dx	=	ó õ p -p  1 2 [sin(m-n)x+sin(m+n)x] dx
   	=	cos[(m-n)x] 2(m-n) - cos[(m+n)x] 2(m+n) ô ô ô p -p
   	=	0.