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Unformatted text preview: HW#10 —Phys374—Spring 2008 Prof. Ted Jacobson Due before class, Friday, April 25, 2008 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374c/ [email protected] 1. Consider the “rectified cosine function” defined by f ( x ) = cos( πx/ 2 L ) , L ≤ x ≤ L, (1) and continued periodically so that f ( x + 2 L ) = f ( x ). [2+3+5+5=15 pts.] (a) Sketch the function f ( x ) over several periods. (b) Use the symmetry to explain why the Fourier coefficients b n vanish. (c) Find the non-vanishing Fourier coefficients. ( Hints : (i) To clean things up, change variables to θ = πx/L . (ii) You’ll need to do a probably unfamiliar integral, which you can look up or work out for yourself.) (d) Using a computer program (Mathematica, Maple, Matlab, or something else) plot the sum of the first few terms in the Fourier series, together with (1), for θ ∈ (- 2 π, 2 π ). Show the result with 1 (just the constant part), 2, 5, and 20 terms included. With 5 terms the sum should already be quite close to (1), except nearincluded....
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