A.2. Notations
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## 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**

- $f=O(g)$ : if $f/g$ is bounded: $|f|\le M|g|$ with some constant $M$;
- $f=o(g)$ : if $f/g\to 0$: $\lim (f/g)=0$;
- $f\sim g$ : if $f/g\to 1$: $\lim (f/g)=1$ which is equivalent to $f=g+o(g)$ or $f=g(1+o(1))$;
- $f \asymp g$ : 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; also $\Omega$ notation (which we do not use).

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