Meaning of $O(n)$ in an expression

Question

As my mathematical knowledge is increasing, I have been seeing more and more of $O(n)$ implementation in expressions. Here is what I mean. Example:
$$z^{q_{N+1} + q_N} w^{q_{N+1} + q_N} (-1)^N (w-1)/w + O(z^{q_{N+1} + q_{N+1}})$$ I understand that $q, N,$ and $w$ are some variables but don't know what to make of $O(z^{q_{N+1} + q_{N+1}})$. How do we consider this in our expression and while we evaluate the statement? I know that this the the Big-O for time complexity of algorithms but am totally confused as to its significance within the expression.

Originally I thought that we would evaluate $z^{q_{N+1} + q_{N+1}}$ while we ignore $O$ but don't know if this makes sense.

Or am I am missing something simple?

Answer 1

It is also known as Landau symbols. In fact there are many such asimptotic symbols, but the most usual of them are $O$ and $o$.

They have meanings either for sequences or for functions.

Sequences.

Let $a_n$ and $b_n$ be two sequences.

$a_n=O(b_n)$ as $n \to \infty$ if $\exists C>0, N=N(C):\, \forall n>N\,|a_n|\leq Cb_n$

$a_n=o(b_n)$ as $n \to \infty$ if $\forall C>0\, \exists N=N(C):\, \forall n>N\,|a_n|\leq Cb_n$

Functions

Let $f(x)$ and $g(x)$ be functions.

$f(x)=O(g(x))$ as $x\to x_0$ if $\exists C>0, \exists \delta=\delta(C): \, |x-x_0|<\delta \Rightarrow|f(x)| \leq Cg(x)$

$f(x)=o(g(x))$ as $x\to x_0$ if $\forall C>0, \exists \delta=\delta(C): \, |x-x_0|<\delta \Rightarrow|f(x)| \leq Cg(x)$

From this it's easy to construct the meaning of the symbols as $x \to -\infty$

When we include the symbol in an expression, we show that there is another member in the expression that however can be negliged in most cases. Also it shows that we don't know the function in question but we do know that it satisfies the inequality above.

Answer 2

Big-O notation is used in asymptotic expansions of an expression that may otherwise not have a simple form - probably most often it's used in power series expansions. The idea is to say that in a given range (usually for large values of x) your original expression $F(x)$ behaves mostly like some other expression $G(x)$, with an error term that behaves like another function $H(x)$, which you would then write as $F(x) = G(x) + O(H(x))$.

So you don't have an exact form of whatever it is you're looking at and you can't use this expression to evaluate it exactly, but you do know that when z (or more likely $|z|$) is "large enough", it behaves mostly like $z^{q_{N+1} + q_N} w^{q_{N+1} + q_N} (-1)^N (w-1)/w$, and the error on treating it that way is approximately some constant times $z^{q_{N+1} + q_{N+1}}$.