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

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

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

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

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

\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

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

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