Closed forms of the Green's function and the generalized Green's function for the Helmholtz operator on the N-dimensional unit sphere

Abstract

The Green's function for the Helmholtz differential operator ∇2 + λ(λ + N − 1) on the N-dimensional (with N ⩾ 1) hyperspherical surface of unit radius is investigated. Its closed form is shown to be where SN is the area of , C(α)λ(x) is the Gegenbauer function of the first kind, while n and n′ are radius vectors, with respect to the centre of , of the observation and source points, respectively. The Green's function G(N)(λ; n, n′) fails to exist whenever λ is such that it holds that λ(λ + N − 1) = L(L + N − 1), with . For these exceptional cases, the generalized (known also as ‘modified’ or ‘reduced’) Green's function is considered. It is shown that may be expressed compactly in terms of the Gegenbauer polynomial C((N−1)/2)L(n ⋅ n′) and the derivative [∂C((N−1)/2)λ(−n ⋅ n′)/∂λ]λ=L. Explicit expressions for the derivatives [∂C(n)λ(x)/∂λ]λ=L and [∂C(n+1/2)λ(x)/∂λ]λ=L, with , are found and used to transform the functions and to potentially more useful forms.