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The first-order ordinary differential equation (ODE) that describes themid-plane gravitational potential in flat finite size discs in which thesurface density is a power-law function of the radius R with exponent s (Hur\'e& Hersant 2007) is solved exactly in terms of infinite series. The formalsolution of the ODE is derived and then converted into a series representationby expanding the elliptic integral of the first kind over its modulus beforeanalytical integration. Inside the disc, the gravitational potential consistsof three terms: a power law of radius R with index 1+s, and two infinite seriesof the variables R and 1/R. The convergence of the series can be accelerated,enabling the construction of reliable approximations. At the lowest-order, thepotential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by avery simple formula whose accuracy (a few percent typically) is easilyincreased by considering successive orders through a recurrence. A basicalgorithm is given. Applications concern all theoretical models and numericalsimulations where the influence of disc gravity must be checked and/or reliablytaken into account.

A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae