The Lommel-Weber Ω function and its application to the problem of electric waves on a thin anchor ring

Abstract

The function Ω n ( x ) defined by the integral 1/ π ∫ π 0 sin ( x sin ϕ — nϕ ) dϕ ), or, when n is an even integer, by 2/ π ∫ π /2 0 sin ( x sin ϕ ) cos nϕdϕ is closely related to J n ( x ), which was first given by Bessel under the form 1/ π ∫ π 0 cos ( x sin ϕ — nϕ ) dϕ . The Ω function, although of less importance in its application than the J function, occurs in a number of problems in mathematical physics, more especially those relating to the interference and diffraction of light.