Find the frequency-dependent susceptibility X () when the temporal susceptibility (t) of a medium is (a) (t ) = 0 (t) (b) (t )

Find the frequency-dependent susceptibility X̂ (ω) when the temporal susceptibility χ(t) of a medium is
(a) χ(t ) = χ₀ δ(t)
(b) χ(t ) = χ₀ θ(t)
(c) χ(t ) = χ₀ θ(t) exp(−t/τ)
(d) χ(t ) = χ₀ θ(t) sin(ω₀t).
Delta functions appear naturally in some of these if you use a convergence factor lim_{ϵ → 0} exp(− t) and remember that lim _{→ 0} ϵ / (ω² + ϵ²) = πδ(ω).