= Angular diameter distance in an open dust universe
{title2=$D_A=S_k(D_C)/(1+z)$}
Normalize the present scale factor to one and assume dust plus curvature with no cosmological constant. Then $H^2=H_0^2[\Omega_Ma^{-3}+(1-\Omega_M)a^{-2}]$ and $D_C=(c/H_0)\int_{1/(1+z)}^1 da/[\sqrt a\sqrt{\Omega_M+(1-\Omega_M)a}]$. For positive $\Omega_K=1-\Omega_M$, the transverse comoving distance is $S_k(D_C)=c\sinh(H_0\sqrt{\Omega_K}D_C/c)/(H_0\sqrt{\Omega_K})$. At fixed radial distance it exceeds the flat transverse distance. At fixed redshift the radial distance must also be calculated from the expansion history. The empty limit is $D_C=(c/H_0)\ln(1+z)$, while the flat dust limit is $2c[1-(1+z)^{-1/2}]/H_0$.
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