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Debaa
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What is an asymptotic function. How do you integrate it?
ThanksSsnow said: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 is a mathematical function that describes the behavior of a function as its input approaches a certain value or infinity. It is used to analyze the growth or decay of a function as its input increases or decreases without bound.
An asymptotic function is typically defined using the big O notation, which describes the upper bound of the growth of a function. It is written as f(x) = O(g(x)) where g(x) is a function that represents the behavior of f(x) as x approaches a certain value or infinity.
Unlike regular functions, which are defined for all values of their input, asymptotic functions are only defined for values approaching a certain value or infinity. They are used to analyze the long-term behavior of a function, rather than its specific values.
Integration of an asymptotic function can be done using the same methods as regular functions, such as the substitution or integration by parts. However, it is important to consider the behavior of the function as its input approaches a certain value or infinity, as it can affect the limits of integration and the resulting integral.
Asymptotic functions are important in mathematics because they allow us to analyze the behavior of functions in the long-term, which can be useful in various fields such as physics, engineering, and economics. They also help us understand the complexity and efficiency of algorithms, as well as make predictions about the behavior of systems or processes.