Fourier- and Laplace Transform

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

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

, where is of course the sign function...

One given explanation is given by "The damping functions and are related by analytic continuation,

"

Since the Laplace Transform is given by

I do not understand how the Fourier Transform has been achieved in a **STRICT MATHEMATICAL WAY WITHOUT HANDWAVING ARGUMENTS**.