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 .