Find the frequency dependent susceptibility X () when the temporal susceptibility (t) of a medium is (a) (t ) 0 (t) (b) (t ) 0 (t) (c) (t ) 0 (t) exp(t ) (d) (t ) 0 (t) sin( 0 t) Delta functions appear naturally in some of these if you use a convergence factor lim 0 exp( t) and remember that lim 0 ...

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

Question:

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).