Gravitation plays an important role in many fields in astrophysics: it appears in the cohesion and stability of bodies such as planets, stars, disks and galaxies. In the Universe, the formation of most astrophysical objects involves disk-like configurations by a main process: the gravitational collapse. The structure and the evolution of these disks (protoplanetary disks, circumplanetary disks...), are an important stage in the process of the formation of stars, planets or satellites. It is therefore fundamental to understand their physics and develop appropriate tools. I devoted my Ph.D. to the computation of the gravitational potential and field of astrophysical disks. Although Newton's force is known for long, the determination of self-gravitating interactions inside bodies remains a difficult task. Strong deviations to sphericity require more efforts. The main difficulty is to manage properly the hyperbolic divergence of the Green kernel 1/(r'-r). In this purpose, the theoretical approach is interesting as it can provide powerful formulae and new tools, which can also help to produce reference solutions. So, I have investigated new methods able to treat this question as rigorously as possible.In a first part, chapter 1 is devoted to the scientifc context and motivations. In the chapter 2 we derive the well known multipole expansion in spherical and cylindrical coordinates from the Poisson equation and Newton's equation. We show the limits of these two developments in the context of astrophysical disks. In chapter 3, we discuss the formalism based on elliptic integrals, its advantages and drawbacks, and we describe two methods which use this approach in the special case of axisymmetrical disks.In the second part, chapter 4 is about the discovery of an alternate formula for the Green kernel, which involves regilar function. To obtain this result, we assume that the disk is vertically homogeneous (i.e., the density varies only with the radius), and that it is axially symetric. In chapter 5, by using this new expression, we build an approximation for the potential in the special case of geometrically thin disks and rings, and another one for systems which are radially confined.In the third part, chapter 6 is devoted to the study of edge effects on the vertical component of the gravitational field caused by a thin disk. According to Paczynski's approximation, the field is a linear function of the surface density pacz78. This approximation is strictly valid only in the infinite slab model, while we are interested in a realistic disk. Close to the outer edges, where gravity decreases, Paczynski's approximation fails and must be corrected. By assuming again a density varying with the radius only, we have derived a new expression for the vertical component of gravitational field, which properly accounts of the presence of the edge of the disk. This is the main subject of the chapter 7. In the last part (chapter 8), we generalize the work by ansorg03, valid under axial symmetry only. Using a similar approach, we built an expression for the self-gravitating potential of cylindrical cells, which is not known in closed form yet. This expression is made of a single integral over the boundary of the cell. This result can be applied in hydrodynamical simulations, where disks are usually discretised into homogeneous cylindrical cells, each cell having its own density.A conclusion and a few perspectives end the thesis.
