In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. It is sometimes expressed in the language of equivalence relations. For example, given complex-valued functions f and g of a natural number variable n, one can write Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Interactive Mathematics Miscellany and Puzzles — A collection of articles on various math topics, with interactive Java... To analyze an algorithm is to determine the amount of resources (such as time and storage) necessary to execute it. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

to express the concept that

.

This defines an equivalence relation; and the equivalence class of f consists of all functions g with similar behaviour to f, in the limit.

Asymptotic notation has been developed to provide a convenient language for the handling of statements about order of growth. It is also called Landau notation, since it became popular first in research in analytic number theory, from about 1900 onwards, introduced by Edmund Landau (originated though by Paul Bachmann). See also Big O notation, for a treatment more from the point of view of analysis of algorithms. The asymptotic point of view is basic in computer science, where the question is typically how to describe the resource implication of scaling-up the size of a computational problem, beyond the 'toy' level. In mathematical analysis, and in particular in the analysis of algorithms, to classify the growth of functions one has recourse to asymptotic notations. ... In complexity theory, computer science, and mathematics, Landau notation, also called little o- and big O notation, is a mathematical notation used for asymptotic comparison of functions. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... Edmund Georg Hermann Landau (February 14, 1877 - February 19, 1938) was a German mathematician and author of over 250 papers on number theory. ... Paul Gustav Heinrich Bachmann (June 22, 1837 - March 31, 1920) was a German mathematician. ... Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ... Flowcharts are often used to represent algorithms. ... Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Downloadable Science and Computer Science books Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: | ...

An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of an infinite series, the partial sums of which do not (necessarily have to) converge; but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide a more and more accurate description of the order of growth of f. An example is Stirling's approximation. In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, Stirlings approximation (or Stirlings formula) is an approximation for large factorials. ...

In symbols, it means we have

but also

and

for each fixed k, while some limit is taken.

In the case that an asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. However, this optimal partial sum will generally have more terms for larger values of the argument.

Asymptotic expansions typically arise in the approximation of certain integrals (saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). For the optimization method called steepest descent see gradient descent. ... For the optimization method called steepest descent see gradient descent. ... The Edgeworth series or Gram-Charlier A series, named in honor of Francis Ysidro Edgeworth, are series that approximate a probability distribution in terms of its cumulants. ...

Further reading

• Erdélyi, A. Asymptotic Expansions. New York: Dover, 1987.