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.


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