# Asymptotic function

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

Ssnow
Gold Member
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

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