FOURIER SERIES FOR FOUR FUNCTIONS & THEIR DERIVATIVES:

Fourier Sine Series

L:=2;

phi := (n,x) -> sin(n*Pi*x/L);

W := x -> 1;

Fourier Series with Weight Function

L := Pi;

phi := (n,x) -> exp(-x)*sin(n*Pi*x/L);

W := x -> exp(2*x);

Initial function "f"

f:= x -> (x^3)*(L-x);

BEGIN PROGRAM RUN WITH SPECIFIED FUNCTION "f"

b:= n -> int(W(x)*f(x)*phi(n,x),x=0..L)/int(W(x)*phi(n,x)^2,x=0..L);

FS:= (N,x) -> sum(b(n)*phi(n,x),n=1..N);

plot({f(x),FS(5,x)},x=0..L,thickness=3);

plot({f(x),FS(10,x)},x=0..L,thickness=3);

plot({f(x),FS(20,x)},x=0..L,thickness=3);

plot({f(x),FS(100,x)},x=0..L,thickness=3);

DFS := (N,x) -> diff(FS(N,x),x);

Df := x -> diff(f(x),x);

plot({Df(x),DFS(5,x)},x=0..L,y=-15..25,thickness=3);

plot({Df(x),DFS(10,x)},x=0..L,y=-8..12,thickness=3);

plot({Df(x),DFS(20,x)},x=0..L,y=-8..10,thickness=3);

plot({Df(x),DFS(100,x)},x=0..L,y=-8..8,thickness=3);

END PROGRAM RUN - SPECIFY NEW "f" & GO BACK UP TO BEGIN PROGRAM RUN

f:= x -> x + Heaviside(x-L/2)*(L-2*x);

f:= x -> Heaviside(x-1);

f:= x -> x^2;