Solid angle subtended by a cylindrical detector at a point source in terms of elliptic integrals

The solid angle subtended by a right circular cylinder at a point source located at an arbitrary position generally consists of a sum of two terms: that defined by the cylindrical surface ($\Omega_{cyl}$) and the other by either of the end circles ($\Omega_{circ}$). We derive an expression for $\Omega_{cyl}$ in terms of elliptic integrals of the first and third kinds and give similar expressions for $\Omega_{circ}$ using integrals of the first and second kinds. These latter can be used alternatively to an expression also in terms of elliptic integrals, due to Philip A. Macklin and included as a footnote in Masket (Rev. Sci. Instr., 28 (3), 191-197, 1957). The solid angle subtended by the whole cylinder when the source is located at an arbitrary location can then be calculated using elliptic integrals.