1. The problem statement, all variables and given/known data The problem is from an optics text, however I believe the problem to be a mathematical one. I'm trying to take the Fourier transform of P(t) = ε0∫ X(t-τ)E(τ) dτ which should equal P(ω) = ε0X(ω)E(ω) where ε0 is a constant X is the susceptibility E is the electric field 2. Relevant equations P(t) = ε0∫ X(t-τ)E(τ) dτ <-starting point Fourier transforms: P(ω) = ∫ P(t)eiωt dt P(t) = 1/2π ∫ F(ω)e-iωt dω P(ω) = ε0X(ω)E(ω) <- end point (final solution) 3. The attempt at a solution First off not sure if convolution is part of the answer where P(t) = ε0∫ X(t-τ)E(τ) dτ also equals P(t) = ε0∫ X(τ)E(t-τ) dτ but I will start my attempt with the first equation If I plug the P(t) time transforms into the left hand side I get 1/2π ∫ F(ω)e-iωt dω = ε0∫ X(t-τ)E(τ) dτ now multiply by eiω'tdt and integrate over all time 1/2π ∫ F(ω)e-iωt dω ∫ eiω'tdt= ε0∫ X(t-τ)E(τ) dτ ∫ eiω'tdt where the left hand side can be arranged to give 1/2π ∫ F(ω) dω ∫ e-i(ω-ω')tdt= ε0∫ X(t-τ)E(τ) dτ ∫ eiω'tdt using ∫ e-i(ω-ω')tdt = 2πσ(ω-ω') <- Dirac Delta function ∫ F(ω)σ(ω-ω') dω = ε0∫ X(t-τ)E(τ) dτ ∫ eiω'tdt <-Dirac Delta function 'picks out' ω' F(ω') = ε0∫ X(t-τ)E(τ) dτ ∫ eiω'tdt and now I'm stuck, unless my approach is wrong to begin with. I also know the electric field is the real part of eiωt so the last equation could be rewritten F(ω') = ε0∫ X(t-τ)eiωτ dτ ∫ eiω'tdt Though I am not sure if this helps.