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.