A new equation for the mid-plane potential of power law discs

Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density $\Sigma(R) \propto R^{s}$ (Huré & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series. Methods. The formal solution of the ODE is derived and then converted into a series representation by expanding the elliptic integral of the first kind over its modulus before analytical integration. Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index $1 + s$, and two infinite series of the variables R and $1/R$. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 ± 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account