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The dependence of the luminosity distance on the redshift has a keyimportance in the cosmology. This dependence can well be given by standardfunctions for the zero cosmological constant. The purpose of this article is topresent such a relation also for the non-zero cosmological constant, if theuniverse is spatially flat. A definite integral is used. The integration endsin the elliptic integral of the first kind. The result shows that no numericalintegration is needed for the non-zero cosmological constant, if the universeis spatially flat.

A curious relation between the flat cosmological model and the elliptic integral of the first kind