Consider the function f: [-, ) R defined by 1, if - x...

 

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Consider the function f: [-π, π) → R defined by 1, if - π ≤ x ≤ πT f(x) 0, if -≤x≤ -1, if < x <π. Denote by g RR the 2π-periodic extension of f to R. (a) Sketch g(x) over the interval x = [-37, 37] indicating carefully the key values on both axes. The Fourier series of f is given by S(x) = lim SN(x) where SN (x) = with 804-N = ao N +Σ (an cos(nx) + b, sin(nx)) n=1 an = 1 |_ f(x) cos(nx) dr πT π and = f(x) sin(nr) dr. TT (b) Give those values of x for which S(x) = f(x) and those values of x for which S(x) = f(x). (c) Determine ao, general expressions for every a and b and show that the Fourier series S of f is S(x) = = 2 πT n=1 (-1)" - cos(") sin(nx). n Consider the function f: [-π, π) → R defined by 1, if - π ≤ x ≤ πT f(x) 0, if -≤x≤ -1, if < x <π. Denote by g RR the 2π-periodic extension of f to R. (a) Sketch g(x) over the interval x = [-37, 37] indicating carefully the key values on both axes. The Fourier series of f is given by S(x) = lim SN(x) where SN (x) = with 804-N = ao N +Σ (an cos(nx) + b, sin(nx)) n=1 an = 1 |_ f(x) cos(nx) dr πT π and = f(x) sin(nr) dr. TT (b) Give those values of x for which S(x) = f(x) and those values of x for which S(x) = f(x). (c) Determine ao, general expressions for every a and b and show that the Fourier series S of f is S(x) = = 2 πT n=1 (-1)" - cos(") sin(nx). n