Define L here L := Pi;

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Define the function over 0 < x < L and zero outside g := x -> piecewise( 0 < x and x < L, 1, 0);

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Extend to an odd function over a period -L < x < L (this step and the next only needed for visualization) h := x -> g(x) - g(-x);

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Extend periodically to two periods each side f:= x-> h(x) + h(x-2*L) + h(x+2*L);

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Set tickmarks for x- and y-axis xmarks := [-2,-1,0,1,2]: ymarks := [-1,0,1]: Graph the function plot(f(x), x=-2*L..2*L, tickmarks = [xmarks,ymarks],scaling = constrained, color = red, thickness = 4, title = "y = f(x)", discont = true);

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N is the max number of terms in partial sum N := 20;

IiM/

b := array(1..N): S := array(1..N): Calculate the coefficients for the Fourier sine series of f(x) for n from 1 to N do b[n] := 2/L * int(f(x)*sin(n*Pi*x/L),x=0..L); if n = 1 then S[n] := b[n] * sin(n*Pi*x/L) else S[n] := S[n-1] + b[n] * sin(n*Pi*x/L) end if; end do: Show result FourierPartialSum:=S[N];

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plot([f(x), S[1]],x = -2*L..2*L, tickmarks=[xmarks,ymarks], color = [red, blue ], thickness = 3, scaling=constrained, discont = true, legend=["f(x)","S_1(x)"]);

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plot([f(x), S[1], S[2]],x = -2*L..2*L, tickmarks=[xmarks,ymarks], color = [red, blue, magenta ], thickness = 3, scaling=constrained, discont = true, legend=["f(x)","S_1(x)","S_2(x)"]);

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plot([f(x), S[1], S[2], S[3]],x = -2*L..2*L, tickmarks=[xmarks,ymarks], color = [red, blue, magenta, black ], thickness = 3,scaling=constrained, discont = true, legend=["f(x)","S_1(x)","S_2(x)","S_3(x)"]);

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with(plots): A := array(1..N): for n from 1 to N do A[n]:= plot([f(x),S[n]],x = -2*L..2*L, tickmarks=[xmarks, ymarks], color = [red,blue], thickness = [4,2], discont = true) end do: display(seq(A[i],i=1..N),insequence=true,scaling=constrained);

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