Hi, I'm attempting to do a fourier transform of using the following definition

I know enough to break the integral up into two separate cases from (-infinity,0) and (0, infinity) and a change of variables for the negative case, but past there I'm stuck.

I have never done integration involving i before and am confused as to what to do. I've tried substitution (x = sqrt(u) , which I can't solve. It has been suggested that I try using the Gaussian function, but I am unsure if it really helps me and I don't know, even if I get an integral that has an anti-derivative that I can use (I know I can convert to a double integral of x,y and convert to polar coordinates, but what would be the value of say

evaluated at r=0 (That's easy 0).. and r=infinity (does the i do anything?, if not then the function diverges and my substitution is useless)

I don't know enough of what do to in this case. I've tried looking online but I can't find a resource that teaches me step by step of what I should be looking for or doing. I tried converting it to a power series and converting it to sines and cosines.

Are there any suggestions on what to do?

Thank you in advance,

Z