I'm reading Classical Electrodynamics 3e by Jackson. In section 1.7 he performs a proof of the Poisson equation in the context of the electric potential. Near the end of the proof, he writes
$$
\nabla^2 \Phi_a(\mathbf{x})
= -\frac{1}{\epsilon_0} \rho(\mathbf{x})\bigl(1 + O(a^2/R^2)\bigr) + O(a^2,a^2\log a) \nabla^2
\rho + \ldots,
$$
where $\rho(\mathbf{x})$ denotes a charge density, $R$ is a radius chosen such that $\rho(\mathbf{x})$ changes little within the sphere centered at $\mathbf{x}$ and bounded by $R$, and $a$ is a variable introduced to make the Laplacian well-behaved.
$O$ denotes big $O$ notation. The meaning of $O(a^2/R^2)$ makes sense to me, but what does $O(a^2,a^2\log a)$ mean?
Best Answer
In Jackson, one can just replace $$O(f(x),g(x))=O(f(x))+O(g(x)).$$ NB: There might be other authors who would use the notation $O(f(x),g(x))$ to mean that it is both $O(f(x))$ and $O(g(x))$.