So the question states:
Find the fourier transform of the function f defined by:
f(x) = -e^(ax) x<0
= e^(-ax) x>0
I think I know what I'm doing, but this is totally new to me, and confirmation that I know what I'm doing would be awesome!!!!
So my answer is:
the transform of e^(-ax) is:
f hat k = integral(0 to infinity) e^(-ax) e^(-ikx) dx
= integral(0 to infinity) e^-(a+ik)x dx
= 1/(a+ik)
the transform of -e^(ax) is:
f hat k = - integral(0 to infinity) e^(ax) e^(-ikx) dx
= - integral(0 to infinity) e^(a-ik)x dx
= - 1/(a-ik)
so total answer is 1/(a+ik) - 1/(a-ik)
So cross multiplying with congugates, gives:
2ik/(a^2+k^2)
Is this correct?
Any help would be hugely appreciated! Thanks!