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Solution to the Massive Laplacian Operator

Let $\phi(x,y)$ be the solution of the PDE: $$(-\Delta + m)\phi(x,y) = \delta(x-y)$$ where $m > 0$ and $\Delta$ is the usual $d$-dimensional Laplace operator. I’ve read that the solution of this equation can be written as: $$\phi(x,y) = \frac{1}{(2\pi)^{d}}\int e^{-ik(x-y)}(k^{2}+m^{2})^{-1}dk$$ but I’m having trouble proving it. The obvious way is to use Fourier […]