Fourier Series

The trigonometric functions {sin(nx), cos(nx)}, n = 0,1,2,..., are useful in creating approximations to functions. Since these functions are all periodic with period , it is convenient to work on the interval [ , ]. The basic idea is to start with a function f ( x ) on [ , ] and attempt to express f ( x ) in the form

where  and  are constants for n  = 0,1,2,... These numbers are referred to as the Fourier coefficients of the function f ( x ).

Orthogonality of cos(nx) and sin(nx)

The calculation of the numbers  and  is based on special properties of integrals involving  and . We list these properties below. Let n  and m  be two distinct  positive integers. Then

The last equality is valid even when m  = n . These equations are referred to as the orthogonality relations  for sines and cosines.

Project - Part 1

Choose several values for m  and n between 1 and 20 and verify the orthogonality relations for sines and cosines.

Square integrals of cos(nx) and sin(nx)

Two other special properties of the functions  and  are given below. Assume n  is an integer greater than or equal to 1.

The special case when n  = 0 is different. In this case we have

 = 0

 =

Project Part 2

Choose several values of n  between 1 and 20 and verify the results above.

A Simple Example

We begin with a simple example. Let .

>	f:=x->4+3*sin(2*x)-5*cos(3*x);

>	plot(f(x),x=-Pi..Pi);

We see what happens when we integrate f ( x ) times  or times .

>	for n from 1 to 5 do int(f(x)*sin(n*x),x=-Pi..Pi) od;

>	for n from 1 to 5 do int(f(x)*cos(n*x),x=-Pi..Pi) od;

 Recall that . Notice that when n  = 2 we have  =  and when n  = 3 we have  = . Finally, we integrate f ( x ) times the constant function 1 (which we think of as cos(0 x )).

>	int(f(x)*1,x=-Pi..Pi);

Project - Part 3

Define the function . Calculate the integrals ,  and  for n  = 1..5. Discuss your results. How are these integrals related to the coefficients of the various sines and cosines in the formula for f ( x )?

Fourier Coefficients

The previous examples suggest that for a (continuous) function f ( x ) defined on [ , ], we define the following Fourier coefficients  and :

, n = 1,2,...

, n  = 1,2,...

These numbers are called the Fourier   coefficients  of f ( x ). Having calculated these coefficients we form the function

 +  +

Theoretically, these sums may involve an infinite number of terms and a question about limits of finite sums arises. For our putposes, we will only consider a finite number of terms in these sums. In most cases 10 terms will be sufficient.

Project - Part 4

Define the function  on the interval [ , ]. Compute the Fourier coefficients of f ( x ) for n  = 0,1,2,...10. It is probably easiest to define  separately and then define the remaining coefficients as functions of the variable n . Also the integrals should be evaluated numerically. For example, one might define the function a  as follows:

>	a:=n->evalf((1/(Pi))*int(f(x)*cos(n*x),x=-Pi..Pi));

>

If you define a  and b  as functions, then you will need to use functional notation a ( n ) in place of subscripts . Also, you can use the above formula for a (0) except that you must multiply the result by 0.5 since a (0) has  in the denominator rather than . After you have defined the Fourier coefficients of f (x), define the partial sum function

Plot the graphs of the original function f ( x ) and  together over the interval [ , ]. How do the graphs compare? What happens if you plot the two graphs on the larger interval [ , ]?

Define a function  of two variables as

This is accomplished with a command of the form

>	S:=(x,k)->.5*a(0)+sum(a(n)*cos(n*x),n = 1 .. k)+sum(b(n)*sin(n*x),n = 1 .. k);

By changing k , you can create different partial sum functions. Find  and  and plot these over the interval [ , ] along with the original function f ( x ). Do the partial sums appear to approximate f ( x ) better as k  increases?

Project - Part 5

We assumed f ( x ) was continuous to guarantee that all the integrals we wrote down existed. These integrals will all exist even if f ( x ) is piecewise continuous over the interval [ , ].

>

restart:

Consider the function f ( x ) defined by -3 over the interval [ ,-2), - x  over the interval [-2,1), and 1 over the interval [1, ]. Such a piecewise defined function can be defined in Maple with the command

>	f:=x->piecewise(x>=-Pi and x<-2,-3,x>=-2 and x<1,-x,x>=1 and x<=Pi,1);

If we evaluate f ( x ) in Maple, we obtain somewhat more familiar output although the inequalities are not exactly in the form we might prefer.

>

f(x);

This function is discontinuous at x  = -2 and at x  = 1.We can plot the graph of this function. Since the function is discontinuous, it is wise to include the option "discont = true" within our plot command.

>	plot(f(x),x=-Pi..Pi,discont=true);

We can work with this function just like any other function. In particular we can compute Fourier coefficients and the partial sum functions just as before. Use the formulas you already have for ,  and . You will  need to copy and execute these formulas if you ran the "restart" command. Calculate  and  and plot these partial sum functions together with f ( x ) over the interval [ , ].