$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ ##A.2. Some notations Compare behaviour of two functions $f(\mathbf{x})$ and $g(\mathbf{x})$ as $\mathbf{x}$ "tends to something" (in the usage it is clear). **Definition 1** 1. $f=O(g)$ : if $f/g$ is bounded: $|f|\le M|g|$ with some constant $M$; 2. $f=o(g)$ : if $f/g\to 0$: $\lim (f/g)=0$; 3. $f\sim g$ : if $f/g\to 1$: $\lim (f/g)=1$ which is equivalent to $f=g+o(g)$ or $f=g(1+o(1))$; 4. $f \asymp g$ : if $f=O(g)$ and $g=O(f)$, which means that $M^{-1}|g|\le f\le M|g|$. We say then that $f$ and $g$ have the same magnitudes. Observe that (2) implies (1) but (1) does not imply (2) and (3) implies (4) but (4) does not imply (3). See in details [Wikipedia](https://en.wikipedia.org/wiki/Big_O_notation); also $\Omega$ notation (which we do not use). _________ [$\Leftarrow$](./SA.1.html)  [$\Uparrow$](../contents.html)