We can extend our notation to the case of two parameters $n$ and $m$ that can go to infinity independently at different rates. For a given function $g(n,m)$, we denote by $O(g(n,m))$ the set of functions

$O(g(n,m)) = \{f(n,m):$ there exist positive constants $c$, $n_0$, and $m_0$ such that $0 \leq f(n,m) \leq cg(n,m)$ for all $n \geq n_0$ or $m \geq m_0 \}$.

Give corresponding definitions for $\Omega(g(n,m))$ and $\Theta(g(n,m))$.

$\Omega(g(n,m)) = \{ f(n,m):$ there exist positive constants $c$, $n_0$, and $m_0$ such that $0 \leq cg(n,m) \leq f(n,m)$ for all $n \geq n_0$ or $m \geq m_0 \}$.

$\Theta(g(n,m)) = \{ f(n,m):$ there exist positive constants $c_1$, $c_2$, $n_0$, and $m_0$ such that $0 \leq c_1g(n,m) \leq f(n,m) \leq c_2g(n,m)$ for all $n \geq n_0$ or $m \geq m_0 \}$.