I was wondering about one (probably trivial) fact during computing the Fourier transform while using contour integral.
As an example I have following function:
$$f(x)={{1}\over{x^2+a^2}}$$ and its Fourier transform is equal to: $$FT[f(x)](s)={\sqrt{\pi\over 2}}e^{-|s|a}.$$
I used this definition of Fourier transform:
$$ FT[f(x)](s) =\frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}f(x){e^{-isx}dx}$$
The singularities are $ia$ and $-ia$ and we have the solutions for $s \lt 0$ and $s \gt 0$. And we are applying the Jordan's lemma.
But why for $s \lt 0$ we integrate along upper half-plane and for $s \gt 0$ we have the lower half-plane? I confuse the sign notation. Everytime I think I have figured it out I have feeling it is not a real explanation, even though the reason is probably pretty simple. But every textbook or internet source just says it is like that and not why.
Let $x=a+bi$ Then we have that $$\exp(-ixs)=\exp(-i(a+bi)s)=\exp(-ias+bs)=\exp(-ias)\exp(bs)$$ If $s \in \mathbb{R}$, then the first term is oscillating. If $s>0$, then $b$ should be negative (otherwise $\exp(bs) \to \infty$) to have a finite integral. If $s<0$, we should have positive $b$. And $b$ is positive in the upper half plane, and negative in the bottom half plane.