An analytical approximation of the luminosity distance in flat
  cosmologies with dark energy
  We present an analytical approximation formula for the luminosity distance in
spatially flat cosmologies with dust and a cosmological constant. We also show
the approximate formulae for the so-called Dyer-Roeder distance (empty beam
case) and the generalised angular diameter distance from redshift $z=z_1$ to
$z=z_2$, which are particularly useful in analysing the gravitational lens
effects. Our formulae are widely applicable over the range of the density
parameter and the redshift with sufficiently small uncertainties. In
particular, in the range of density parameter $0.3 \lid \Omega_{\rmn m} \lid 1$
and redshift $0.03 \lid z \lid 1000$, the relative error for the luminosity
distance by our formula is always smaller than that of the recent work by
\cite{wu}.
