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Motivated by the inverse problem for the Lemaitre-Tolman-Bondi dust solution,in which problem the luminosity distance function $D_L(z)$ is taken as an inputto select a specific model, we compute the function $D_L(z)$ of the LTBsolution up to the third order of $z$. To perform the otherwise cumbersomecomputation, we introduce a new convenient form of the LTB solution, in whichthe solution is explicit and unified. With this form of the LTB solution weobtain the luminosity distance function with the full generality. We inparticular find that the function exactly coincides with that of a homogeneousand isotropic dust solution up to second order, if we demand that the solutionbe regular at the center.Comment: 18 pages, 2 figures, including 2 new paragraphs and 1 figure in  section 2, final version to appear in Classical and Quantum Gravit

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