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I'm reading Phillips & Panofsky's textbook on Electromagnetism: Classical Electricity and Magnetism. At chapter 14, section 2, we are presented with a solution of the wave equations for the potentials through Fourier Analysis. Eventually, the authors arrive at an equation for the Green function for the Helmholtz Equation:

$\nabla^2G(x_{\alpha},x_{\alpha}') + \frac{\omega^2}{c^2}G(x_{\alpha},x_{\alpha}') = - \delta(x_{\alpha} - x_{\alpha}')$.

In here, one can understand $x_{\alpha}$ simply as the position vector in the unprimed coordinates (where the field is being measured) and $x_{\alpha}'$ as the primed coordinates (where the integrals are being evalueated).

P&P solve the equation for $r = |x_{\alpha} - x_{\alpha}'| > 0$ and in doing this they find $G(r) = \frac{A}{r}e^{\pm i k r}$, in which $k = \frac{\omega}{c}$. With this equation in hands, they claim that the value of $A$ might be obtained by integrating the original PDE for $G$ in a volume with very small $r$, such that $G$ behaves as $\frac{A}{r}$. Then they obtain that $- 4 \pi A = -1$ and thus $A = \frac{1}{4\pi}$, as it should be.

When calculating this integral, what happens with the integral of $k^2 G$? This should diverge, if I'm correct, but they seem to simply pretend this term is not there at all and forget about it. What is happening and why can we ignore this integral? I looked for some references on finding the Green functions for the Helmholtz Equation, but neither of them solved this problem, and they usually they solved the equation for $G$ through othr methods.

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They're not just integrating with respect to $r$, they're integrating over the volume of a small sphere, so \begin{align} \lim_{R\rightarrow0^+}\int_0^R k^2 G 4\pi r^2 \operatorname{d}r & =\lim_{R\rightarrow0^+}\int_0^R 4\pi A \int_0^R r + \mathcal{O}(r^2)\operatorname{d}r \\ &=\lim_{R\rightarrow0^+} 4\pi A \left[\frac{R^2}{2} + \mathcal{O}(R^3)\right]\\ &\rightarrow0 \end{align}

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