There is a closed-form solution of the light-travel distance if Ω Ω r = Ω Ω m = 0 involving the inverse hyperbolic functions arcosh or arsinh (or involving inverse trigonometric functions if the cosmological constant has the other sign). If Ω Ω r = Ω Ω Λ Λ = 0 then there is a closed-form solution for d T ( z ) but not for z ( d T ) .