High School Asymptotic Function: Definition & Integration

[Debaa]

What is an asymptotic function. How do you integrate it?

 

[Ssnow]

An asymptotic function $f$ is a function that has the same behaviour of another function $g$, at least in a small neighborhood ...
For example the Taylor expansion gives you a polynomial that has the same behaviour of $g$. The Taylor expansion is not always practicable. In mathematics there is a notation used in the asymptotic expansion called ''big-$O$'' notation.
For discrete functions $f(n)=O(g(n))$ if $g$ is an upper bound on $f$: there exists a fixed constant $c$ and a fixed $n_{0}$ such that for all $n≥n_{0}$,

$f(n) ≤ cg(n)$.

We say $f$ is $o(g(n))$ (read: "$f$ is little-$o$ of $g$'') if for all arbitrarily small real $c > 0$, for all but perhaps finitely many $n$,

$f(n) ≤ cg(n)$.

We say that f is $\Theta(g(n))$ (read: "$f$ is theta of $g$") if $g$ is an accurate characterization of $f$ for large $n$: it can be scaled so it is both an upper and a lower bound of $f$.

Details of Taylor expansion, $O$-notation, or asymptotic analysis are in https://en.wikipedia.org/wiki/Taylor_series , https://en.wikipedia.org/wiki/Big_O_notation , https://en.wikipedia.org/wiki/Asymptotic_analysis

Ssnow

 

[Debaa]

An asymptotic function $f$ is a function that has the same behaviour of another function $g$, at least in a small neighborhood ...
For example the Taylor expansion gives you a polynomial that has the same behaviour of $g$. The Taylor expansion is not always practicable. In mathematics there is a notation used in the asymptotic expansion called ''big-$O$'' notation.
For discrete functions $f(n)=O(g(n))$ if $g$ is an upper bound on $f$: there exists a fixed constant $c$ and a fixed $n_{0}$ such that for all $n≥n_{0}$,

$f(n) ≤ cg(n)$.

We say $f$ is $o(g(n))$ (read: "$f$ is little-$o$ of $g$'') if for all arbitrarily small real $c > 0$, for all but perhaps finitely many $n$,

$f(n) ≤ cg(n)$.

We say that f is $\Theta(g(n))$ (read: "$f$ is theta of $g$") if $g$ is an accurate characterization of $f$ for large $n$: it can be scaled so it is both an upper and a lower bound of $f$.

Details of Taylor expansion, $O$-notation, or asymptotic analysis are in https://en.wikipedia.org/wiki/Taylor_series , https://en.wikipedia.org/wiki/Big_O_notation , https://en.wikipedia.org/wiki/Asymptotic_analysis

Ssnow

Thanks