Q) Use the residue theorem and contour integration to show that the inverse Fourier transform of the function

is given by the function

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My current strategy is to us the formula for the inverse Fourier transform

and integrate about a closed contour around the pole at , which would then equal times the residue at that point ( ). This would then equal the integral along the real axis between -R and R, plus the integral along the open path given by the semi circle under the real axis which encircles the pole at -j. This all gives (I hope) the following

My question is, am I going about this the right way? I've attempted to follow the method described above, but it leads to what seems to be a dead end (I get a horrific integral with exponentials of exponentials)...

Any help would be much appreciated!