The same periodicity properties as described in above apply to G α α β β . Specifically, G α α β β ( τ τ ∣ ∣ τ τ ′ ) = G α α β β ( τ τ − − τ τ ′ ) and G α α β β ( τ τ ) = G α α β β ( τ τ + β β ) , for τ τ < 0 .