Example 2.1.1
Compute the Fourier cosine series for , on ( ).  Create a plot that shows the function  and the partial sum of the first 20 terms of the Fourier cosine series on the interval .

First code the orthonormal system, the inner product on ( ), and the function .

>	phi[0]:=1/sqrt(Pi); phi[n]:=sqrt(2/Pi)*cos(n*x); dp:=(f,g)->int(f*g,x=0..Pi); f:=1+x;

Create the Fourier constants, also called Fourier coefficients.

>	c[0]:=dp(f, phi[0]); c[n]:=dp(f, phi[n]);

>	ps:=unapply(simplify(c[0]*phi[0])+'sum'(c[n]*phi[n], n=1..m), (m,x));

>	p1:=plot(ps(20,x), x=-3*Pi..3*Pi): p2:=plot(f, x=0..Pi, color=green, thickness=4): display([p1, p2], scaling=constrained, view=[-3*Pi..3*Pi, 0..5]);

>

Example 2.1.2
Compute the Fourier sine series for , on ( ).  Create a plot that shows the function  and the partial sum of the first 50 terms of the Fourier sine series on the interval .

First code the orthonormal system, the inner product on ( ), and the function .

>	phi[n]:=sqrt(2/Pi)*sin(n*x); dp:=(f,g)->int(f*g,x=0..Pi); f:=1+x;

Create the Fourier coefficients.

>	c[n]:=dp(f, phi[n]);

>	ps:=unapply('sum'(c[n]*phi[n], n=1..m), (m,x));

>	p1:=plot(ps(50,x), x=-3*Pi..3*Pi): p2:=plot(f, x=0..Pi, color=green, thickness=4): display([p1, p2], scaling=constrained, view=[-3*Pi..3*Pi, -5..5]);

>

Example 2.1.3
Let    on ( ) and , on [ ) .  Compute the Fourier series for , and create a plot that shows the function  and the partial sum of the first 50 terms of the Fourier  series on the interval .

First code the orthonormal system, the inner product on ( ), and the function .

>	phi[0]:=1/sqrt(2*Pi); phi[2*n]:=1/sqrt(Pi)*sin(n*x); phi[2*n-1]:=1/sqrt(Pi)*cos(n*x); dp:=(f,g)->int(f*g,x=-Pi..Pi); f:=piecewise(-Pi<x and x<0, 1, 0<=x and x<Pi, 1+x, undefined);

Create the Fourier coefficients.

>	c[0]:=dp(f,phi[0]); c[2*n]:=dp(f, phi[2*n]); c[2*n-1]:=dp(f, phi[2*n-1]);

>	ps:=unapply(simplify(c[0]*phi[0])+'sum'(c[2*n]*phi[2*n]+c[2*n-1]*phi[2*n-1], n=1..m), (m,x));

>	p1:=plot(ps(50,x), x=-3*Pi..3*Pi): p2:=plot(f, x=-Pi..Pi, color=green, thickness=4): display([p1, p2], scaling=constrained, view=[-3*Pi..3*Pi, 0..5]);

>

Example 2.1.4 (Problem 4, Section28, of the Textbook)
Find the Fourier series for , on ( ), where .  Use Euler's formula  to write
.  Then, take  and create a plot that shows the function  and the partial sum of the first 100 terms of the Fourier  series on the interval .

Note that I used  instead of the book's .  This is necessary in order to avoid programming confusion between the  in  and the  in the coefficient .  Technically,  is an element of the table  with alphanumeric index .

First code the orthogonal  system , the inner product on ( ), and the function .

>	phi[0]:=1; phi[n]:=exp(I*n*x); dp:=(f,g)->int(f*g,x=-Pi..Pi); assume(alpha, real); f:=exp(alpha*x);

Create the Fourier coefficients.  Note that, since the system  is not normalized, an appropriate multiplicative constant  needs to be introduced for the computation of the Fourier coefficients, much like the formulas for  and   we derived in the beginning of the course.  

>	c[0]:=1/Pi*dp(f,phi[0]); c[n]:=1/Pi*dp(f, phi[n]);

Next, derive the formulas for  and , where  is greater than or equal to .

>	a[0]:=combine(convert(c[0], trig), trig); a[n]:=combine(convert(Re(c[n]), trig), trig); b[n]:=combine(convert(Im(c[n]), trig), trig);

>	ps:=unapply(simplify(a[0]/2)+'sum'(map(factor, a[n]*cos(n*x)+b[n]*sin(n*x)), n=1..m), (m, x));

>	p1:=plot(subs(alpha=1, ps(100, x)), x=-3*Pi..3*Pi): p2:=plot(subs(alpha=1, f), x=-Pi..Pi, color=green, thickness=4): display([p1, p2], view=[-3*Pi..3*Pi, -1..exp(Pi)]);

>