(2)O O O (1)O 9.3 Application Fourier Series of Piecewise Smooth Functions (p608)To define piecewise functions you can use either piecewise (do ?piecewise for the syntax) or use the book's suggestion, which is to define a function "unit" that is one on the interval (a,b) and zero otherwise:unit := (t,a,b) -> Heaviside(t-a) - Heaviside(t-b);unit:=t,a,b/HeavisidetKaKHeavisidetKbplot(unit(t,0.5,1.5),t=-2..2);K2K1 0 1 20.20.40.60.81.0Here is the function in Figure 9.3.8. This function is even, thus to find its Fourier coefficients we only need to define it on [0,Pi]:f := t -> (Pi/3)*unit(t, 0, Pi/6) + (Pi/2 - t)*unit(t, Pi/6, 5*Pi/6) + (-Pi/3)*unit(t, 5*Pi/6, Pi);f:=t/13 p unit t, 0,16 p C12 pKt unit t,16 p,56 pK13 p unit t,56 p, pTo double check that we have the correct expression we can plot the functionplot(f(t),t=0..Pi);(3)O O (5)O (4)1 2 3K1.0K0.500.51.0Since the function is EVEN b_n = 0 for n>=1, thus we only compute the a_n:a := n -> (2/Pi) * int(f(t)*cos(n*t),t=0..Pi);a:=n/2 0pf t cosn tdtpThe "assuming integer" at the end of the expression tells Maple that n is an integer for addition simplifications, however an expression does not come to mind!a(n) assuming integer;2 cos16 n pKcos56 n pp n2So we simply compute the Fourier sum, with say 25 termsfouriersum := a(0)/2 + sum(a(n)*cos(n*t),n=1..25);fouriersum:=2 cost 3pK225 3 cos 5 tpK249 3 cos 7 tp(5)O C2121 3 cos 11 tpC2169 3 cos 13 tpK2289 3 cos 17 tpK2361 3 cos 19 tpC2529 3 cos 23 tpC2625 3 cos 25 tpAnd we check that the Fourier sum is consistent with the function we asked.plot(fouriersum, t= -2*Pi..3*Pi);K6K4K202 4 6 8K1.0K0.50.51.0In the same way it is possible to compute the coefficients b(n) for odd functions or more generally the a(n) and b(n) for general (not odd neither even)
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