Series Expansions for the Solid Angle Subtended by a Circular Disk at a Point Directly Above the Periphery

Abstract

It is the purpose of this paper to establish a new series development representing the solid angle subtended by a circular disk at a point situated directly above and in the immediate vicinity of the periphery. After having recalled Masket's series expansion which is applicable only for distances larger than the diameter of the disk, a few initial terms of the new series expansion are deduced starting from an integral representation of the solid angle and applying some formulas from the theory of Legendre's complete elliptic integrals. In this way, it is shown that the series must contain logarithmic terms besides integer powers of the distance variable. Then, after having transformed the problem under study into an equivalent problem of electrostatics, the complete infinite series representation of the solid angle is deduced from its expression as a Fourier integral by applying a new expansion theorem to the latter. The paper ends with an interesting verification of the obtained result.