The European Space Agency mission Gaia has published with its second data release (DR2) a catalogue of photometric measurements for more than 1.3 billion astronomical objects in three passbands. The precision of the measurements in these passbands, denoted G, G_BP, and G_RP, reaches down to the milli-magnitude level. The scientific exploitation of this data set requires precise knowledge on the response curves of the three passbands. This work aims to improve the exploitation of the photometric data by deriving an improved set of response curves for the three passbands, allowing for an accurate computation of synthetic Gaia photometry. This is achieved by formulating the problem of passband determination in a functional analytic formalism, and linking the photometric measurements with four observational, one empirical and one theoretical spectral library. We present response curves for G, G_BP, and G_RP that differ from the previously published curves, and which provide a better agreement between synthetic Gaia photometry and Gaia observations.