Fourier Approxiamtion for the function f(x) where f(x) is periodic (with period 2*Pi) and a period is defined as f(x) = -7 between -Pi and 0 and f(x) = 7 between 0 and Pi

restart:with(plots):

f:=x->piecewise(x>-Pi and x < 0, -7, x>0 and x<Pi, 7);

plot(f(x),x=-2*Pi..2*Pi);

Let's look at calculating the first few fourier coefficients

a_0:=1/(2*Pi)*int(f(x),x=-Pi..Pi);

a_1:=1/Pi*int(f(x)*cos(x),x=-Pi..Pi);b_1:=1/Pi*int(f(x)*sin(x),x=-Pi..Pi);

a_0+a_1*cos(x)+b_1*sin(x);

plot({f(x),28*sin(x)/Pi},x=-Pi..Pi);

a_2:=1/Pi*int(f(x)*cos(2*x),x=-Pi..Pi);b_2:=1/Pi*int(f(x)*sin(2*x),x=-Pi..Pi);a_3:=1/Pi*int(f(x)*cos(3*x),x=-Pi..Pi);b_3:=1/Pi*int(f(x)*sin(3*x),x=-Pi..Pi);

a_0+a_1*cos(x)+b_1*sin(x)+a_2*cos(2*x)+b_2*sin(2*x)+a_3*cos(3*x)+b_3*sin(3*x);

plot({f(x),28*sin(x)/Pi+28/3*sin(3*x)/Pi},x=-Pi..Pi);

plot({f(x),(28/Pi)*sin(x)+(28/(3*Pi))*sin(3*x)+(28/(5*Pi))*sin(5*x)+(28/(7*Pi))*sin(7*x)+(28/(9*Pi))*sin(9*x)},x=-Pi..2,y=-10..10);

Here are some commands so that the coefficients can be written as sequences

restart:

f:=x->piecewise(x>-Pi and x < 0, -7, x>0 and x<Pi, 7);

L:=2*Pi;a_0:=int(f(x),x=-L/2..L/2)/L;

a:=(m,L)->2*int(cos(2*m*Pi*x/L)*f(x),x=-L/2..L/2)/L; seq(a[p]=a(p,L),p=1..5);

b:=(m,L)->2*int(sin(2*m*Pi*x/L)*f(x),x=-L/2..L/2)/L; seq(b[p]=b(p,L),p=1..5);

Now let's look at some plots as we add more fourier coefficients

a:=(m,L)->2*int(cos(2*m*Pi*x/L)*f(x),x=-L/2..L/2)/L; seq(a[p]=a(p,L),p=1..21);

b:=(m,L)->2*int(sin(2*m*Pi*x/L)*f(x),x=-L/2..L/2)/L; seq(b[p]=b(p,L),p=1..21);

plot({f(x),28*sin(x)/Pi+28*sin(3*x)/(3*Pi)+28*sin(5*x)/(5*Pi)+28*sin(7*x)/(7*Pi)+28*sin(9*x)/(9*Pi)+28*sin(11*x)/(11*Pi)+28*sin(13*x)/(13*Pi)+28*sin(15*x)/(15*Pi)},x = -Pi .. Pi,y = -10 .. 10);

plot({f(x),28*sin(x)/Pi+28*sin(3*x)/(3*Pi)+28*sin(5*x)/(5*Pi)+28*sin(7*x)/(7*Pi)+28*sin(9*x)/(9*Pi)+28*sin(11*x)/(11*Pi)+28*sin(13*x)/(13*Pi)+28*sin(15*x)/(15*Pi)+28*sin(17*x)/(17*Pi)+28*sin(19*x)/(19*Pi)+28*sin(21*x)/(21*Pi)},x = -Pi .. Pi,y = -10 .. 10);

And now a quick animation of the convergence

with(plots):

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