Asymptotic analysis

In mathematics and its applications, asymptotic analysis is a method to the boundary behavior of functions classified by describing only the main trend of the boundary behavior.

Description of the asymptotic behavior

The asymptotic behavior of functions can be described by an equivalence relation. If, for example, and real-valued functions of natural numbers n, so can be an equivalence relation by defining

If and only if

The equivalence class of consists of all functions for which the relative error to the border crossing tends to 0. This definition can be applied to functions of a real or complex variable and, in the case against, the approach to often is only a subset, for example, in the real domain from the left or from the right, or in the complex in an angular range immediately or over a specified discrete set.

Some examples of asymptotic results

• The prime number theorem of number theory states that the number of primes less than for large asymptotically behaves like.
• The Stirling formula describes the asymptotic behavior of the faculties.
• Four elementary examples, and with the asymptotic behavior, and tends to 0 for

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 is the Landau notation complexity theory, be tested in the asymptotic runtime and memory consumption of an algorithm.

The simplest way to define these symbols is the following: and are classes of functions with the properties

Is usually clear from the context. Next one often writes instead of the following:.

Asymptotic Expansion

Under an asymptotic expansion of a function is defined as the representation of the function as not necessarily convergent series. The partial sums of such a series does not need to converge, but provide near good approximations for the function value. The best known example of an asymptotic expansion is Stirling's series as asymptotic expansion for the faculty. Define can be such a development with the help of an asymptotic sequence as

With.

If the asymptotic expansion does not converge, there is for each function argument an index in which the approximation error

Is the smallest; Adding more terms worsens the approximation. The index of the best approximation is in asymptotic expansions however, the greater is closer to.

Asymptotic developments occur particularly in the approximation of certain integrals, for example by means of the saddle point method. The asymptotic behavior of rows it can be often traced with the help of Euler's summation formula.