I have a memory-friction kernel (yes its physics, but still a math question) defined as:

$\displaystyle \gamma (t) = \Theta (t)\cos(\omega_{\sigma}t)$

The Fourier Transform is given as (without explanation, except the term retarded, which causes the $\displaystyle -i\epsilon$):

$\displaystyle \tilde{\gamma}(\omega) = \lim\limits_{\epsilon\to 0^+}\frac{-i \omega}{\omega_{\sigma}^2 - \omega^2 - i\epsilon\, sgn(\omega)}$

, where $\displaystyle sgn(x)$ is of course the sign function...
One given explanation is given by "The damping functions $\displaystyle \hat{\gamma}(z)$ and $\displaystyle \tilde{\gamma}(\omega)$ are related by analytic continuation,

$\displaystyle \hat{\gamma}(z) = \tilde{\gamma}(\omega=iz);\quad \tilde{\gamma}(\omega) = \lim\limits_{\epsilon\to 0^+}\hat{\gamma}(z=-i\omega + \epsilon)$"

Since the Laplace Transform is given by

$\displaystyle \hat{\gamma}(z) = \frac{z}{\omega_{\sigma}^2 + z^2}$

I do not understand how the Fourier Transform $\displaystyle \tilde{\gamma}(\omega)$ has been achieved in a STRICT MATHEMATICAL WAY WITHOUT HANDWAVING ARGUMENTS.