Fourier Transform Problem

I have got the below fourier problem:

The Fourier transform of a function $\displaystyle f(t)$ is defined as:

$\displaystyle F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt$

I have worked through it and got two different answers depending on whether I used a positive contour or negative contour.

For the positive contour:

$\displaystyle j\sqrt{\frac{\pi}{2}} e^\omega$

For the negative contour:

$\displaystyle j\sqrt{\frac{\pi}{2}} e^{-\omega}$

I'm not sure whether these need to be combined or which one to use for the final answer. Looked at Wolfram Alpha and this gave something completely different.

Can someone please point me in the right direction.