Asymptotic sequence

In analysis , an asymptotic sequence is a basic building block of an asymptotic analysis . The asymptotic sequence defines the starting space of an asymptotic development and thus determines the possible results of the analysis.

definition

A finite or infinite sequence of functions in the domain is called asymptotic for if ${\displaystyle (\phi _{n})}$ ${\displaystyle \Omega }$ ${\displaystyle x\to x_{0}}$

${\displaystyle \phi _{n+1}(x)={\hbox{o}}(\phi _{n}(x)),\;x\to x_{0}}$,

with the Landau notation . In the case of infinite sequences, one speaks of a uniform asymptotic sequence in n , if uniformly applies in n, or of a uniform asymptotic sequence in the parameters , if the sequence depends on a parameter and applies uniformly in the parameters. ${\displaystyle \phi _{n+1}={\hbox{o}}(\phi _{n})}$ ${\displaystyle \epsilon }$ ${\displaystyle \phi _{n+1}={\hbox{o}}(\phi _{n})}$

Examples

• The sequence of real functions for . ${\displaystyle \phi _{n}=x^{n}}$ ${\displaystyle x\to 0}$
• The sequence of real functions with for . ${\displaystyle \phi _{n}=x^{-\lambda _{n}}}$ ${\displaystyle \lambda _{n+1}>\lambda _{n}}$ ${\displaystyle x\to \infty }$

properties

A partial sequence of an asymptotic sequence is also asymptotic, and raising the complete sequence to the power of a positive number again yields an asymptotic sequence.

literature

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