Someone told me this week that the Landau notation is very pratical in general in analysis.
Definition : Let the function $\phi$ defined on an open set containing $x_0$.We want to compare $f$ à $\phi$: we want to know if $\displaystyle{\left\vert\frac{f}{\phi}\right\vert}$ is bound or if the limit is nul at this point; but, we can write the rapport only if $ \phi$ doesn't vanish.
We say that $ f\in o(\phi)$ in the neighborhood of $ x_0$ if and only if for all $ \varepsilon >0$ there exists $ \eta>0$ such that
$ \displaystyle{\forall x\in ]x_0-\eta,x_0+\eta[, \vert f(x)\vert<\varepsilon \vert\phi(x)\vert}$
If $ \phi$ doesn't vanish, we have : $ \displaystyle{\forall x\in ]x_0-\eta,x_0+\eta[, \left\vert\frac{f(x)}{\phi(x)}\right\vert<\varepsilon }$ We say in this case that $ f$ is negligible in front of $ \phi$ in a neighborhood of $ x_0$.
I would like to know why it is important to embrace this notation and master the concept. Does someone could explain to me the importance of this notation (With examples, comparative, etc.)?
The question may be silly, but sometimes I wonder about certain topic without being myself able to answer this question. I'm a little young (13 years old), and still have a lot to learn in mathematics.
Thanks!
This is useful in the field of algorithmic analysis. For example, if you want to calculate $\exp x$, you can use its definition, $$ \exp x = \sum_{j=0}^\infty \frac{x^j}{j!} $$ but unfortunately, computers cannot perform infinite number of operations within a finite amount of time, so it must be truncated to yield a polynomial, $$ \exp x \simeq \sum_{j=0}^{m-1} \frac{x^j}{j!} = 1 + x + \frac{x^2}{2} + \cdots + \frac{x^{m-1}}{(m-1)!} $$ Now here is the problem, where to choose $m$? Each term in this polynomial adds significant computation time. We may want to control the error within a certain bound. Let's say that we want the error to be below $x^2$ in the interval $]-1,1[$. Since $x^3 \in o(x^2), x^4 \in o(x^2), \dots$, $$ \exp x = 1 + x + \frac{x^2}{2} + o(x^2) $$ That is, we only need to calculate $3$ terms so that the error is bounded below some multiple of $x^2$.
As in mathematical analysis, say if you want to find the limit of this function $$ \lim_{x \to \infty} \frac{1}{1 + \frac{1}{x} + \frac{1}{x^2}} $$ Since $\frac{1}{x} \in o(1), \frac{1}{x^2} \in o(1)$, the above limit is equal to $$ \lim_{x \to \infty} \frac{1}{1} = 1 $$