Green's Function--Helmholtz Differential Equation

The inhomogeneous Helmholtz differential equation is

(1)

where the Helmholtz operator is defined as . The Green's function is then defined by

(2)

Define the basis functions as the solutions to the homogeneous Helmholtz differential equation

(3)

The Green's function can then be expanded in terms of the s,

(4)

and the delta function as

(5)

Plugging (◇) and (◇) into (◇) gives

(6)

Using (◇) gives

(7)

(8)

This equation must hold true for each , so

(9)

(10)

and (◇) can be written

(11)

The general solution to (◇) is therefore

			(12)
			(13)