Luminosity distance-redshift relation for the LTB solution near the centre

Motivated by the inverse problem for the LemaItre-Tolman-Bondi dust solution in which problem the luminosity distance function DL(z) is taken as an input to select a specific model, we compute the function DL(z) of the LTB solution up to the third order of z. To perform the otherwise cumbersome computation, we introduce a new convenient form of the LTB solution, in which the solution is explicit and unified. With this form of the LTB solution we obtain the luminosity distance function with full generality. We, in particular, find that the function exactly coincides with that of a homogeneous and isotropic dust solution up to second order, if we demand that the solution be regular at the centre