# What is the general solution to the two-dimensional inhomogeneous Helmholtz equation

The general solution to the equation:

$$\nabla^2 u[x,y,z]+k^2 u[x,y,z]=\rho [x,y,z]$$

is known in terms of an integral over $$\rho$$, eg here eqn 1258. What is the solution for the equivalent 2D problem:

$$\nabla^2 u[x,y]+k^2 u[x,y]=\rho [x,y]$$

The general solution is $$u(\mathbf{x}) =u_0(\mathbf{x})- \frac{i}{4}\int_{\mathbb R^2}H_0^{(1)}(k|\mathbf{x}-\mathbf{x}'|)\rho(\mathbf{x}')d^2\mathbf{x}',$$ where $$u_0$$ is any solution to the homogeneous equation and $$H_0^{(1)}$$ is a Hankel function.