The positive and negative frequency parts of F are probably not the same as they would be if you did a Fourier decomposition of F without transforming to the w-plane.
I don't know why he thought this is a smiley-easy exercise, he seems sometimes totally out of touch with what a beginner would have at their fingertips. Unless he thinks people can shake a little QFT onto the page by waving their fingers. I guess after decades of living it, QFT evaporates off your skin. Your mind turns tea into QFT.