Im working through Zee and I'm having a little trouble with some integrals. I'm trying to reproduce the analogue of the inverse square law for a 2+1 D universe and I figured I could start with the statement for the energy $$E=-\int \frac{d^2 k}{(2 \pi)^2}\frac{e^{i \vec{k}\cdot (\vec{x}-\vec{y})}}{k^2 + m^2}=-\int_{0}^{\infty}\frac{k\,dk}{(2\pi)^2(k^2+m^2)}\int_{0}^{2\pi}e^{ik|x-y|\cos\phi}d\phi$$ which turns into $$=-\int_{0}^{\infty}\frac{k\,J_{0}(k|x-y|)\, dk}{2\pi (k^2+m^2)}$$ Then contour integrated after seeing $k$ and $J_0$ were both odd to get $$-\frac{(2\pi i)(im)J_{0}(im|x-y|)}{4\pi (2im)}=-\frac{i\, J_{0}(im|x-y|)}{4}$$ but this is imaginary, where did I go wrong?
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