The subject of Fourier series is concerned with the representation of a 2-periodic

Chapter 5, Problem 35

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The subject of Fourier series is concerned with the representation of a 2-periodic function f as thefollowing infinite linear combination of the set offunctions{1,sin nx, cos nx}n=1 :f (x) = 12a0 +n=1(an cos nx + bn sin nx). (5.2.5)In this problem, we investigate the possibility of performingsuch a representation.(a) Use appropriate trigonometric identities, or someform of technology, to verify that the set of functions{1,sin nx, cos nx}n=1is orthogonal on the interval [,].(b) By multiplying (5.2.5) by cos mx and integratingover the interval [,], show thata0 = 1 f (x) dxandam = 1 f (x) cos mx dx.[Hint: You may assume that interchange of theinfinite summation with the integral is permissible.](c) Use a similar procedure to show thatbm = 1 f (x)sin mx dx.It can be shown that if f is in C1(,), thenEquation (5.2.5) holds for each x (,). Theseries appearing on the right-hand side of (5.2.5)is called the Fourier series of f , and the constantsin the summation are called the Fouriercoefficients for f .(d) Show that the Fourier coefficients for the functionf (x) = x, < x , f (x + 2) = f (x),arean = 0, n = 0, 1, 2,...bn = 2n cos n, n = 1, 2,...and thereby determine the Fourier series of f .(e) Using some form of technology, sketch theapproximations to f (x) = x on the interval(,) obtained by considering the first threeterms, first five terms, and first ten terms in theFourier series for f . What do you conclude?