Unformatted text preview: = 1 2 ² 2 π ´ sin 2( w + 1) π w + 1 + sin 2( w1) π w1 µ ( w ± = 1) = 1 2 ² 2 π ´ sin 2 πw w + 1 + sin 2 πw w1 µ ( w ± = 1) = ² 2 π sin2 πw w w 21 ( w ± = 1) . Also, by l’Hospital’s rule, we have lim w → ² 2 π sin 2 πw w w 21 = √ 2 π, which is the value of the cosine transform at w = 1. To write f as an inverse Fourier cosine transform, we appeal to (6). We have, for x > 0, 2 π ³ ∞ w w 21 sin 2 πw cos wxdw = ± cos x if 0 < x < 2 π , i f x > 2 π. For x = 2 π , the integral converges to 1 / 2. So 2 π ³ ∞ w w 21 sin2 πw cos 2 πwdw = 1 2 ....
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 Fall '11
 StuartChalk
 Fourier cosine transform

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