I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed). e.g.
- $X = O_k(Y)$ (or $X \ll_k Y$, or $Y \gg_k X$) means that $|X| \leq C_k Y$ for some $C_k$ depending only on $k$.
- $X = o_{n \to \infty; k}(Y)$ means that $|X| \leq c_k(n) Y$ for some function $c_k(n)$ of both $k$ and $n$, which goes to zero as $n \to \infty$ for fixed $k$.
- (Rarer) $X = O_{n \to \infty; k}(Y)$ means that $|X| \geq C_k Y$ whenever $n \geq N_k$, for some $C_k$ and $N_k$ depending only on $k$.
It can be instructive to rewrite some basic notions in analysis in this sort of notation, just to get a slightly different perspective. For instance, if $f: {\bf R} \to {\bf R}$ is a function, then:
- $f$ is continuous iff one has $f(y) = f(x) + o_{|y-x| \to 0; f,x}(1)$ for all $x,y \in {\bf R}$
- $f$ is uniformly continuous iff one has $f(y) = f(x) + o_{|y-x| \to 0; f}(1)$ for all $x,y \in {\bf R}$
- A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is equicontinuous if one has $f_n(y) = f_n(x) + o_{|y-x| \to 0; F,x}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$ (note that the implied constant depends on the family $F$, but not on the specific function $f_n$ or on the index $n$)
- A sequence $F = (f_n)_{n \in {\bf N}}$ of functions is uniformly equicontinuous if one has $f_n(y) = f_n(x) + o_{|y-x| \to 0; F}(1)$ for all $x,y \in {\bf R}$ and $n \in {\bf N}$
- $f$ is differentiable iff one has $f(y) = f(x) + (y-x) f'(x) + o_{|y-x| \to 0; f,x}(|y-x|)$;
- (similarly for uniformly differentiable, equidifferentiable, etc.)
(These formulations are close to the nonstandard analysis formulations of these concepts, which also uses similar but not quite identical asymptotic notation, but that is another story.)