Dispersion relations of free electron laser(FEL) are analysed. By solving the FEL′s pendulum equation, gain coefficient Im(n) and refractive index Re(n)-1 are devived, then it is proved that there are Hilbert transformative relations between Re(n)-1 and Im(n), which are called dispersion relations, or Kramers Kronig relations. Finally, the physical analyses on dispersion relations are made.