A.1. Field theory

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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

  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.

Obviously (c) implies (d) but (d) does not imply (c).

See in details Wikipedia; also $\Omega$ notation (which we do not use).

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