An analytical approximation of the luminosity distance in flat
  cosmologies with a cosmological constant
  We present an analytical approximation formula for the luminosity distance in
spatially flat cosmologies with dust and a cosmological constant. Apart from
the overall factor, the effect of non-zero cosmological constant in our formula
is written simply in terms of a rational function. We also show the approximate
formulae for the Dyer-Roeder distance (empty beam case) and the generalized
angular diameter distance from redshift $z_1$ to $z_2$, which are particularly
useful in analyzing 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 $0.3 \leq \Omega_{\rm m} \leq 1$ and redshift $0.03 \leq z \leq
1000$, the relative error for the luminosity distance by our formula is always
smaller than that of the recent work by Wickramasinghe and Ukwatta (2010).
Hence, we hope that our formulae will be an efficient and useful tool for
exploring various problems in observational cosmology.
