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Unformatted text preview: MA 2930, April 6, 2011 Worksheet 10 1. Let f ( x ) = cos 2 x, x / 2 (a) Extend f ( x ) to an even periodic function and find its Fourier series. (b) Extend f ( x ) to an odd periodic function and find its Fourier series. Where do the series converge and to what? (a) If you draw the graph of f ( x ) extended to an even function (by reflecting the portion from 0 to / 2 in the y-axis) and then to a periodic function, youll see that the resulting function is cos 2 x . And the Fourier series of cos 2 x is clearly itself since its already a linear combination of sines and cosines of appropriate frequencies. (b) If you draw the graph of f ( x ) extended to an odd function (by re- flecting it in the origin) and then to a periodic function, youll see that its actually a periodic function of period / 2 equal to f ( x ) on (0 ,/ 2). Since the function is odd, the Fourier series has only sine terms whose coefficients are b n = 1 / 4 Z / 2 cos 2 x sin( nx / 4 ) dx = 4 Z / 2 cos 2 x sin 4 nxdx = 2 Z / 2 sin(4 n + 2) x + sin(4 n...
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