# Asymptotic analysis

In mathematics and its applications called asymptotic analysis around the one way limiting behavior of functions to be classified by describing only the major trend of marginal behavior.

## Description of the asymptotic behavior

The asymptotic behavior of functions can be described with an equivalence relation . Let and be real-valued functions of natural numbers n , then an equivalence relation can be defined by ${\ displaystyle f}$${\ displaystyle g}$

${\ displaystyle f \ sim g}$

exactly when

${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {f (n)} {g (n)}} = 1}$

applies. The equivalence class of consists of all functions for which the relative error to the border crossing tends to 0th This definition can be directly transferred to functions of a real or complex variable as well as to the case against , whereby the approximation to often only takes place via a subset, e.g. B. in the real from the left or from the right, or in the complex in an angular range, or over a predetermined discrete set. ${\ displaystyle g}$${\ displaystyle h}$ ${\ displaystyle {\ tfrac {h (n) -g (n)} {g (n)}}}$${\ displaystyle g}$${\ displaystyle n \ to \ infty}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x_ {0}}$${\ displaystyle x_ {0}}$

## Some examples of asymptotic results

• The prime number theorem of number theory says that the number of prime numbers smaller for large behaves asymptotically like .${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x / \ log (x)}$
• The Stirling formula describes the asymptotic behavior of the faculties .
• Four elementary examples are , , and with the asymptotic behavior , , or for up to 0th${\ displaystyle \ ln (1 + x)}$${\ displaystyle \ sin (x)}$${\ displaystyle 1- \ cos (x)}$${\ displaystyle \ cot (x)}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x ^ {2} / 2}$${\ displaystyle 1 / x}$${\ displaystyle x}$

## Landau notation

A useful notation for describing the growth classes is the Landau notation, which originally comes from Paul Bachmann , but was made famous by Edmund Landau . An important application of the Landau notation is complexity theory , in which the asymptotic runtime and memory consumption of an algorithm are examined.

The simplest way to define these symbols is as follows: and are classes of functions with the properties ${\ displaystyle O (f (x))}$${\ displaystyle o (f (x))}$

For all ,${\ displaystyle g (x) \ in O (f (x)) \ Leftrightarrow \ limsup \ limits _ {x \ to x_ {0}} \ left | {\ frac {g (x)} {f (x)} } \ right | <\ infty}$
For everyone .${\ displaystyle g (x) \ in o (f (x)) \ Leftrightarrow \ lim \ limits _ {x \ to x_ {0}} \ left | {\ frac {g (x)} {f (x)} } \ right | = 0}$

The point usually becomes clear from the context. Next one often writes instead of the following: . ${\ displaystyle x_ {0}}$${\ displaystyle g (x) \ in O (f (x))}$${\ displaystyle g (x) = O (f (x))}$

## Asymptotic development

Under an asymptotic expansion of a function refers to the representation of the function as formal power series - so as not necessarily convergent series . In this case, after a finite term has been terminated, the size of the error term can be checked, so that the asymptotic expansion provides a good approximation in the vicinity of for the function value . A well-known example of an asymptotic expansion is the Stirling series as an asymptotic expansion for the faculty . Such a development can be defined with the help of an asymptotic sequence as ${\ displaystyle f}$${\ displaystyle x_ {0}}$${\ displaystyle f (x_ {0})}$ ${\ displaystyle (\ varphi _ {n})}$

${\ displaystyle f (x) = \ sum _ {i = 1} ^ {N} a_ {i} \ varphi _ {i} (x) + o (\ varphi _ {N} (x))}$

with . ${\ displaystyle \ varphi _ {n + 1} (x) = o (\ varphi _ {n} (x)), \; x \ to x_ {0}}$

If the asymptotic expansion does not converge, there is an index for each function argument for which the approximation error ${\ displaystyle x}$${\ displaystyle k}$

${\ displaystyle f (x) - \ sum _ {i = 1} ^ {k} a_ {i} \ varphi _ {i} (x)}$

becomes smallest in terms of amount; Adding more terms worsens the approximation. The index of the best approximation is in asymptotic expansions but the bigger the closer in lies. ${\ displaystyle k}$${\ displaystyle x}$${\ displaystyle x_ {0}}$

Asymptotic developments occur in particular when approximating certain integrals, for example using the saddle point method . The asymptotic behavior of series can often be traced back to this with the help of Euler's empirical formula .

## literature

• A. Erdélyi: Asymptotic Expansions. Dover Books on Mathematics, New York 1987, ISBN 0-486-60318-0 .
• L. Berg: Asymptotic Representations and Developments. German Publishing House of Science, Berlin 1968, DNB 750308605 .

## Individual evidence

1. asymptotic development of a function . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .