λ λ is purely imaginary if ε ε ( ω ω ) is real and β β 2 ε ε ( ω ω ) > 1 . That is, when ε ε ( ω ω ) is real, Cherenkov radiation has the condition that v > c ε ε ( ω ω ) = c n . This is the statement that the speed of the particle must be larger than the phase velocity of electromagnetic fields in the medium at frequency ω ω in order to have Cherenkov radiation. With this purely imaginary λ λ condition, λ λ ∗ ∗ / λ λ = i and the integral can be simplified to: ( d E d x particle ) rad = q 2 c 2 ∫ ∫ ε ε ( ω ω ) > 1 β β 2 ω ω ( 1 − − 1 β β 2 ε ε ( ω ω ) ) d ω ω = q 2 c 2 ∫ ∫ v > c n ( ω ω ) ω ω ( 1 − − c 2 v 2 n 2 ( ω ω ) ) d ω ω