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The Fourier transform $X(j \omega)$ of the signal

$$x(t) = \frac{t}{(1+t^{2})^{2}}$$

is ______________.

  1. $\frac{\pi}{2j} \omega e^{– |\omega|}$
  2. $\frac{\pi}{2} \omega e^{– |\omega|}$
  3. $\frac{\pi}{2j}e^{– |\omega|}$
  4. $\frac{\pi}{2}e^{– |\omega|}$
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