Consider a periodic function f(x) with period 2 f(x) = and satisfying the periodicity condition f(x+2) = f(x) for all x outside this range. i) Sketch a graph of the function in the range -3π ≤ x ≤ 3. ii) Now consider expanding f(x) in the periodic basis as ao f(x) 2 e 6 2. Now use the formula = to show that where the coefficients {a} and {a} need to be determined. 1. Using symmetry arguments alone show that ak = 0 V k. (ii) +[ã sin(kx) + ak cos(kx)], k=1 ak = = = f(x) cos(kx) dx, Using the result of part (b) show that f(x) defined in the region - ≤ x ≤ π as = x² 8WI 8WI k=1 k² 4(-1)k k² 이념 || -cos(kx) + and 킨 π² +7. 3 12 Turn over 5/9

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