Product of Big-O Estimates
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Theorem
Sequences
Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.
Let:
- $a_n = \map \OO {b_n}$
- $c_n = \map \OO {d_n}$
where $\OO$ denotes big-$\OO$ notation.
Then:
- $a_n c_n = \map \OO {b_n d_n}$
Real Analysis
Let $c$ be a real number.
Let $f, g : \hointr c \infty \to \R$ be real functions.
Let $\OO$ denote big-$\OO$ notation.
Let $R_1 : \hointr c \infty \to \R$ be a real function such that $f = \map \OO {R_1}$.
Let $R_2 : \hointr c \infty \to \R$ be a real function such that $g = \map \OO {R_2}$.
Then:
- $f g = \map \OO {R_1 R_2}$