# 2011_Tutorial_5 - n-1 x 2 n-1 4 Find the Fourier series of...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 5 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

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Unformatted text preview: n-1) x 2 n-1 4. Find the Fourier series of a half-wave rectiﬁer u ( t ) = (-π/w < t < sin wt < t < π/w and u ( t + 2 π w ) = u ( t ). Ans . 1 π + 1 2 sin wt-2 π ( 1 1 · 3 cos2 wt + 1 3 · 5 cos4 wt + ··· ) MA1505 Tutorial 5 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 2...
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