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I have been trying to find the (causal) Green's function of

[tex] \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + m^2 \phi = 0. [/tex]

What would be a good way to approach this? I have initial values for t=0, so I use Laplace transforms on t and Fourier transforms for the spatial variables. However, I end up with a divergent integral involving the Bessel function, and I haven't been able to find a contour on which the integral converges. Has anyone seen this before, and what would be a good way to approach this equation?

Note that we could simply take the Green's function for the 3-d Helmholtz to solve this equation, but it wouldn't be causal. Is there something conceptual, that would allow me to go directly from the 3-d solution to the 2-d causal Green's function?

Thanks!

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# Green's function of Helmholtz eqn (with time)

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