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Tutorial_4 - u t Ans 1 π 1 2 sin wt-2 π 1 1 3 cos2 wt 1...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 4 1. Show that the Fourier series for f ( x ) = 1 2 ( x + | x | ) , - π < x < π ; f ( x + 2 π ) = f ( x ) is given by π 4 + X 1 ( - 1) n - 1 πn 2 cos nx + ( - 1) n +1 n sin nx . 2. Find the Fourier series that represent the following graph: Ans . 3 2 - 2 π X n =1 sin(2 n - 1) x 2 n - 1 3. Sketch the graph of the following periodic function. Determine whether it is an even or odd function, and find its Fourier series. f ( x ) = ( 1 - π/ 2 < x < π/ 2 0 π/ 2 < x < 3 π/ 2 , f ( x + 2 π ) = f ( x ) . Ans . 1 2 + 2 π X n =1 ( - 1) n +1 cos(2 n - 1) x 2 n - 1 4. Find the Fourier series of a half-wave rectifier u ( t ) = ( 0 - π/w < t < 0 sin wt 0 < t < π/w and u ( t + 2 π w ) =
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Unformatted text preview: u ( t ). Ans . 1 π + 1 2 sin wt-2 π ( 1 1 · 3 cos2 wt + 1 3 · 5 cos4 wt + ··· ) MA1505 Tutorial 4 5. Find the Fourier series for the function f ( x ) = (-2-x-2 < x < 2-x < x < 2 and f ( x + 4) = f ( x ). Ans . 4 π ∞ X n =1 1 n sin nπx 2 6. Find the sine and cosine Fourier half range expansion for f ( x ) = x, < x < π. Also, sketch the corresponding expansions of f . Ans . ∞ X n =1 (-1) n +1 2sin nx n ; π 2 + ∞ X n =1 2((-1) n-1) π cos nx n 2 2...
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