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I was reading a text about quantum scattering, and I faced a notation I don't understand. The equation is the following: $$ \nabla \psi_{\text{scattered}} = \frac{i k f(\theta) e^{ikr}}{r} \mathbf{\hat{r}} + \mathcal{O}\left(\frac{1}{r^2}\right) $$ and I do not know the meaning of the $\mathcal{O}\left(\frac{1}{r^2}\right)$ portion.

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It is a common notation in computer science, mostly used in asymptotic complexity. The definition can be found here.

Informally, it means that the quantity $\nabla \psi_{\text{scattered}}$, when $r$ goes to infinity, is equal to $\frac{ikf(\theta)e^{ikr}}{r}$ plus some unknown function $g$ that does not grow quicker than the function $r \mapsto \frac{1}{r^2}$.

From another viewing point, for very large $r$ (the very large is essential), if you approximate $\nabla \psi_{\text{scattered}}$ with $\frac{ikf(\theta)e^{ikr}}{r}$, you know that the error will not grow faster than $\frac{1}{r^2}$.

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  • $\begingroup$ Thanks Nelimee. Your answer was really helpful:) $\endgroup$ – Audrey Sep 10 '19 at 12:09