Legendre’s well-known elliptic integrals are not the only version of elliptic integrals. Carlson’s form, developed in the late 1970s, have many advantages, and are particularly well suited for Hertzian contact analysis. They fit immediately into the basic formulation: they make no distinction between the major and minor axes of the ellipse (reducing the number of equations needed): and the extension to the study of the deformation outside the contact area is barely noticeable: nothing like the switch from complete to incomplete integrals needed when using Legendre’s integrals is required. And finally, their computation is rapid and straightforward. In addition, equations as Carlson integrals are given for the displacements due to tangential loading (Cattaneo–Mindlin theory), and notes given on the elliptic integrals needed in the evaluation of the internal stresses in a Hertzian contact.