Helmholtz Equation - Inhomogeneous Helmholtz Equation

The inhomogeneous Helmholtz equation is the equation

where ƒ : Rn → C is a given function with compact support, and n = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the k term) is switched to a minus sign.

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

uniformly in with, where the vertical bars denote the Euclidean norm.

With this condition, the solution to the inhomogeneous Helmholtz equation is the convolution

(notice this integral is actually over a finite region, since has compact support). Here, is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with ƒ equaling the Dirac delta function, so G satisfies

The expression for the Green's function depends on the dimension of the space. One has

for n = 1,

for n = 2, where is a Hankel function, and

for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for .