restart: with(plots): # Basis functions for Fourier series: C := (x,n) -> cos(2*Pi*n*x/L); S := (x,n) -> sin(2*Pi*n*x/L); E := (x,n) -> exp(2*Pi*n*I*x/L); L := 1; # Period of series # Example coefficients for a particular square-wave function: a := n -> (6/n)*sin(4*n*Pi/3); b := n -> (6/n)*(1-cos(4*n*Pi/3)); # Define coefficient values for n=0 case separately a(0) := 1/2; b(0) := 0; # Fourier coefficients of complex exponential series for a real-valued function: c := n -> (a(n)-I*b(n))/2; c(0) := a(0); # Fourier sums \342\200\223 general forms of sine/cosine or exponential series # Note use of "add" instead of "sum" [ask me] FS := (x,N) -> add( a(i)*C(x,i) + b(i)*S(x,i),i=0..N); FSE:= (x,N) -> add( c(i)*E(x,i),i=-N..N); # Gibbs phenomenon [look it up!] is clear in plot of square wave plot(FS(x,30),x=-2..2); # Same function but plotted using exponential series. Take Re part (even though it's a real function) so Maple isn't confused by machine-error sized imaginary parts plot(Re(FSE(x,30)),x=-2..2); JSFH