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 I'm now trying to determine the fundamental solution to the Helmholtz wave equation. My starting point is $$(\nabla^2+k^2)G(\vec{r})=-4\pi\delta(\vec{r})$$. Treating this as a distributional statement, it is equivalent to $$\int G(\vec{r})(\nabla^2+k^2)e^{-2\pi i\vec{\xi}\cdot\vec{r}}d^3\vec{r}=-4\pi$$ This leads to the desired result after some contour integration. But if I want to be a little more rigorous and not rely on distributional techniques, I run into trouble. Integrating both sides over punctured $\textbf{R}^3$: \begin{align*} 0&=\lim_{\epsilon\to 0}\int_{B^c_\epsilon}e^{-2\pi i\vec{\xi}\cdot\vec{r}}(\nabla^2+k^2)G(\vec{r})d^3\vec{r}\\ &=\lim_{\epsilon\to 0}\int_{B^c_\epsilon}\left[\nabla^2\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)-G(\vec{r})\nabla^2e^{-2\pi i\vec{\xi}\cdot\vec{r}}-2\nabla G(\vec{r})\cdot\nabla e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right]d^3\vec{r}+k^2\tilde{G}(\vec{\xi})\\ &=\left[4\pi^2\xi^2+k^2\right]\tilde{G}(\vec{\xi})+\lim_{\epsilon\to 0}\left\{ \int_{\partial B^c_\epsilon}\nabla\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)\cdot d\vec{A}+4\pi i\vec{\xi}\cdot\int_{B^c_\epsilon}e^{-2\pi i\vec{\xi}\cdot\vec{r}}\nabla G(\vec{r})d^3\vec{r}\right\}\\ &=\left[k^2-4\pi^2\xi^2\right]\tilde{G}(\vec{\xi})+\lim_{\epsilon\to 0}\left\{\int_{\partial B^c_\epsilon}\nabla\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)\cdot d\vec{A}+4\pi i\vec{\xi}\cdot\int_{\partial B^c_\epsilon}G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}d\vec{A}\right\} \end{align*} The problem is that I want the remaining integrals to evaluate to a constant, but it appears that it will depend on $$\lim_{\epsilon\to 0}G(\epsilon)$$ and $$\lim_{\epsilon\to 0}\left.\frac{\partial G}{\partial r}\eval\right|_{r=\epsilon}$$ I can solve the integrals by assuming spherical symmetry of G(r), but I just end up with functions of $\xi$, $\epsilon$ and G, and it is unclear if there is even convergence in the $\epsilon\to 0$ limit. How can I proceed from this point? EDIT: I just had a thought. We are dealing with a second order PDE, so there are two degrees of freedom to the solution. Could we choose G(0) and G'(0) in such a way as to get reduce the remaining integrals to a normalization constant?