Use the Parseval relation to evaluate (a) J -00 (b) / do (w + a) w...

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Use the Parseval relation to evaluate (a) J -00 (b) / do (w² + a²)²¹ w² dw (w² + a²)² (a) Show that the Fourier sine and cosine transforms of e-atare 2 2 a • √ √²= 6² + = πως ta Tw² + a²¹ 8c (w): Hint. Each of the transforms can be related to the other by integration by parts. 8s (w) = (b) Show that @sin wx w² + a² 0 -do cos cx w² + a² 1=1/√e-ax dw= Л - e-ar 2a x > 0, x > 0. Use the Parseval relation to evaluate (a) J -00 (b) / do (w² + a²)²¹ w² dw (w² + a²)² (a) Show that the Fourier sine and cosine transforms of e-atare 2 2 a • √ √²= 6² + = πως ta Tw² + a²¹ 8c (w): Hint. Each of the transforms can be related to the other by integration by parts. 8s (w) = (b) Show that @sin wx w² + a² 0 -do cos cx w² + a² 1=1/√e-ax dw= Л - e-ar 2a x > 0, x > 0.

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